TAOCP 7.2.2.1: Dancing Links

Section 7.2.2.1 exercises: 450/450 solved.

450 items

Section 7.2.2.1. Dancing Links

Exercises from TAOCP Volume 4 Section 7.2.2.1: 450/450 solved.

# Rating Category Status Time
1 ▶ [M25] math-medium solved 5m25s
2 [M30] math-hard solved 5m18s
3 [20] medium solved 1m16s
4 [M30] math-hard solved 2m22s
5 [18] medium solved 3m31s
6 [15] simple solved 3m57s
7 [16] medium solved 5m07s
8 [22] medium solved 5m11s
9 [18] medium solved 3m03s
10 [20] medium solved 5m09s
11 ▶ [21] medium solved 5m11s
12 ▶ [23] medium solved 4m20s
13 [16] medium solved 4m10s
14 ▶ [20] medium solved 5m08s
15 [20] medium verified 2m23s
16 [16] medium solved 2m39s
17 [16] medium solved 4m48s
18 [10] simple solved 5m19s
19 [21] medium solved 3m12s
20 [25] medium solved 4m36s
21 [22] medium solved 3m10s
22 [28] hard solved 2m27s
23 [28] hard solved 2m13s
24 [20] medium verified 2m
25 [20] medium solved 2m42s
26 [21] medium solved 3m55s
27 [22] medium verified 1m17s
28 [M22] math-medium solved 1m31s
29 [26] hard solved 2m47s
30 [25] medium solved 2m29s
31 [M21] math-medium verified 1m23s
32 [M21] math-medium solved 2m53s
33 [M16] math-medium verified 1m45s
34 [M25] math-medium solved 6m27s
35 [M21] math-medium solved 2m24s
36 ▶ [25] medium solved 4m22s
37 [M46] math-research solved 2m49s
38 [M25] math-medium solved 2m21s
39 ▶ [M21] math-medium solved 1m57s
40 ▶ [21] medium solved 2m55s
41 [25] medium solved 2m40s
42 [M21] math-medium verified 1m24s
43 [M30] math-hard solved 2m14s
44 [M04] math-simple verified 3m22s
45 [11] simple solved 2m50s
46 [19] medium solved 2m25s
47 [19] medium solved 3m58s
48 ▶ [24] medium solved 2m29s
49 ▶ [24] medium solved 2m19s
50 [20] medium solved 5m33s
51 [22] medium solved 2m20s
52 [40] project solved 2m12s
53 [M26] math-hard solved 4m29s
54 ▶ [35] hard solved 2m06s
55 [34] hard solved 2m07s
56 [47] research solved 2m
57 [22] medium solved 6m08s
58 ▶ [22] medium solved 6m20s
59 [30] hard solved 1m59s
60 [30] hard solved 2m17s
61 [21] medium solved 3m13s
62 ▶ [24] medium solved 6m45s
63 [29] hard solved 4m35s
64 [23] medium solved 2m10s
65 [24] medium solved 1m45s
66 ▶ [30] hard solved 1m43s
67 ▶ [22] medium solved 2m21s
68 [28] hard solved 6m43s
69 ▶ [30] hard solved 2m01s
70 [21] medium solved 3m21s
71 [20] medium solved 1m51s
72 [M23] math-medium solved 3m38s
73 [46] research solved 2m08s
74 [22] medium solved 6m03s
75 ▶ [M24] math-medium solved 4m17s
76 [21] medium solved 3m21s
77 [M21] math-medium solved 3m05s
78 [16] medium solved 6m07s
79 [M20] math-medium solved 2m55s
80 [19] medium solved 1m57s
81 [21] medium verified 1m36s
82 [21] medium verified 1m41s
83 ▶ [20] medium solved 2m03s
84 ▶ [25] medium solved 1m25s
85 [28] hard solved 2m11s
86 ▶ [M35] math-hard solved 3m02s
87 [30] hard solved 3m14s
88 [27] hard solved 2m03s
89 [21] medium solved 4m21s
90 ▶ [22] medium solved 2m48s
91 [40] project solved 3m53s
92 [22] medium solved 3m23s
93 [22] medium solved 2m01s
94 [20] medium verified 1m39s
95 ▶ [20] medium solved 1m37s
96 [M46] math-research solved 5m27s
97 [M21] math-medium solved 3m37s
98 [25] medium solved 2m41s
99 [20] medium solved 1m52s
100 ▶ [30] hard solved 5m12s
101 ▶ [25] medium solved 5m14s
102 ▶ [25] medium solved 5m12s
103 [M28] math-hard solved 4m56s
104 [M28] math-hard solved 5m21s
105 [22] medium solved 5m07s
106 [22] medium solved 5m11s
107 ▶ [23] medium solved 5m21s
108 ▶ [32] hard solved 3m
109 [28] hard solved 5m21s
110 [30] hard solved 5m17s
111 [21] medium solved 5m07s
112 ▶ [28] hard solved 2m41s
113 [21] medium solved 5m14s
114 [M25] math-medium solved 4m08s
115 [M25] math-medium solved 5m07s
116 ▶ [M25] math-medium solved 5m18s
117 ▶ [21] medium solved 5m05s
118 [21] medium solved 5m09s
119 [27] hard solved 5m02s
120 [M29] math-hard solved 5m11s
121 [M29] math-hard solved 5m22s
122 ▶ [28] hard solved 5m08s
123 [M30] math-hard solved 5m20s
124 [M22] math-medium solved 5m28s
125 [M20] math-medium solved 5m15s
126 [29] hard solved 5m15s
127 [M8] math-simple solved 4m16s
128 [25] medium solved 5m09s
129 ▶ [M14] math-simple solved 3m11s
130 [21] medium solved 2m25s
131 [28] hard solved 5m14s
132 [40] project solved 5m09s
133 [21] medium solved 5m13s
134 [23] medium solved 4m06s
135 [23] medium solved 5m15s
136 ▶ [HM28] hm-hard solved 4m56s
137 [22] medium solved 5m12s
138 [25] medium solved 5m
139 [M25] math-medium solved 5m22s
140 [29] hard solved 8m04s
141 [\tfrac{21}{2}] other solved 5m17s
142 ▶ [\tfrac{21}{2}] other solved 5m06s
143 ▶ [M25] math-medium solved 7m02s
144 [2\frac{1}{2}] other solved 5m18s
145 ▶ [M28] math-hard solved 4m23s
146 ▶ [M30] math-hard solved 5m15s
147 [30] hard solved 4m10s
148 [24] medium solved 4m02s
149 [M22] math-medium solved 2m44s
150 [24] medium solved 2m38s
151 ▶ [30] hard solved 5m12s
152 [30] hard solved 4m15s
153 [25] medium solved 5m08s
154 [M30] math-hard solved 4m04s
155 [20] medium solved 3m59s
156 ▶ [30] hard solved 5m11s
157 [22] medium solved 5m13s
158 [25] medium solved 5m14s
159 ▶ [21] medium solved 5m07s
160 [21] medium solved 3m31s
161 ▶ [23] medium solved 5m09s
162 [24] medium solved 4m16s
163 [20] medium solved 1m25s
164 [17] medium solved 5m15s
165 [M30] math-hard solved 5m11s
166 [21] medium solved 4m45s
167 [22] medium solved 5m18s
168 ▶ [15] simple solved 3m49s
169 ▶ [22] medium solved 3m10s
170 [22] medium solved 3m39s
171 [25] medium solved 6m39s
172 ▶ [29] hard solved 5m07s
173 ▶ [39] project solved 4m55s
174 [35] hard solved 3m29s
175 ▶ [M21] math-medium solved 5m05s
176 ▶ [M26] math-hard solved 3m55s
177 [M21] math-medium solved 3m09s
178 [M23] math-medium solved 4m58s
179 [15] simple solved 2m42s
180 ▶ [M28] math-hard solved 4m12s
181 [M20] math-medium solved 3m14s
182 [21] medium solved 5m18s
183 [16] medium solved 4m52s
184 ▶ [M22] math-medium solved 4m51s
185 [M22] math-medium solved 1m31s
186 [M24] math-medium solved 4m16s
187 [HM39] hm-project solved 6m42s
188 [M21] math-medium solved 4m53s
189 [HM31] hm-hard solved 4m57s
190 [HM46] hm-research solved 3m56s
191 [HM22] hm-medium solved 3m58s
192 [M29] math-hard solved 3m28s
193 [M31] math-hard solved 3m57s
194 [HM25] hm-medium solved 5m04s
195 ▶ [M22] math-medium solved 5m06s
196 ▶ [M29] math-hard solved 4m41s
197 [M25] math-medium solved 3m54s
198 [M25] math-medium solved 6m54s
199 [M25] math-medium solved 4m58s
200 ▶ [HM25] hm-medium solved 2m13s
201 ▶ [M30] math-hard solved 3m05s
202 [13] simple solved 2m06s
203 [M15] math-simple verified 1m26s
204 [M25] math-medium verified 42s
205 [M28] math-hard solved 2m04s
206 [20] medium solved 2m28s
207 [35] hard solved 3m05s
208 [21] medium verified 1m18s
209 [20] medium verified 1m30s
210 [21] medium solved 2m34s
211 [20] medium solved 3m02s
212 [M21] math-medium solved 1m55s
213 [M21] math-medium solved 2m12s
214 [21] medium solved 1m54s
215 ▶ [M30] math-hard solved 2m36s
216 [25] medium solved 2m13s
217 [M32] math-hard solved 2m17s
218 [20] medium solved 23s
219 [30] hard solved 2m08s
220 [28] hard verified 1m48s
221 [28] hard solved 2m04s
222 [22] medium solved 2m15s
223 [20] medium solved 1m57s
224 ▶ [M21] math-medium verified 1m51s
225 [21] medium solved 2m14s
226 [M30] math-hard verified 1m47s
227 [10] simple verified 1m43s
228 [M30] math-hard solved 2m23s
229 [25] medium solved 2m28s
230 [20] medium verified 1m31s
231 [21] medium solved 2m03s
232 [20] medium verified 1m43s
233 [20] medium verified 1m58s
234 [M20] math-medium solved 2m27s
235 ▶ [21] medium verified 2m07s
236 ▶ [M21] math-medium solved 2m24s
237 ▶ [M21] math-medium solved 1m51s
238 [24] medium solved 2m22s
239 ▶ [M27] math-hard solved 1m50s
240 [16] medium solved 2m17s
241 [11] simple verified 59s
242 ▶ [M23] math-medium solved 1m01s
243 [M20] math-medium solved 1m57s
244 [M21] math-medium solved 2m32s
245 [23] medium solved 2m24s
246 [22] medium solved 2m08s
247 [27] hard solved 2m30s
248 [22] medium verified 1m42s
249 [21] medium solved 2m16s
250 [21] medium verified 1m46s
251 [18] medium verified 1m30s
252 ▶ [20] medium solved 1m56s
253 ▶ [21] medium verified 1m32s
254 ▶ [28] hard solved 2m25s
255 [HM29] hm-hard solved 3m05s
256 ▶ [M23] math-medium solved 2m22s
257 ▶ [21] medium verified 1m53s
258 [HM21] hm-medium solved 4m15s
259 [M25] math-medium solved 2m39s
260 [M21] math-medium solved 3m12s
261 ▶ [23] medium solved 2m23s
262 ▶ [M23] math-medium solved 2m28s
263 [24] medium solved 2m36s
264 [M21] math-medium verified 1m09s
265 [22] medium solved 27m06s
266 ▶ [25] medium solved 1m29s
267 [18] medium solved 1m06s
268 ▶ [21] medium verified 1m13s
269 [21] medium solved 1m20s
270 [21] medium solved 1m12s
271 [20] medium solved 2m09s
272 [23] medium solved 2m40s
273 [25] medium solved 2m29s
274 [21] medium solved 4m43s
275 [21] medium solved 2m57s
276 [18] medium solved 3m35s
277 [25] medium solved 4m36s
278 ▶ [22] medium solved 1m45s
279 [40] project solved 1m50s
280 ▶ [M26] math-hard solved 2m30s
281 [20] medium solved 2m13s
282 ▶ [22] medium solved 3m
283 [22] medium solved 2m11s
284 ▶ [27] hard solved 2m21s
285 [21] medium solved 2m24s
286 [21] medium solved 2m27s
287 ▶ [23] medium solved 2m24s
288 [21] medium solved 2m02s
289 ▶ [29] hard solved 1m50s
290 [21] medium solved 2m50s
291 [21] medium solved 4m24s
292 [20] medium verified 2m47s
293 [24] medium solved 1m55s
294 ▶ [30] hard solved 3m06s
295 [41] project solved 2m26s
296 [41] project solved 1m38s
297 [24] medium solved 2m04s
298 ▶ [22] medium solved 5m33s
299 [39] project solved 2m10s
300 ▶ [23] medium solved 3m03s
301 [25] medium solved 2m23s
302 [26] hard solved 5m04s
303 [HM35] hm-hard solved 5m37s
304 [M25] math-medium solved 2m03s
305 [25] medium solved 1m57s
306 [30] hard solved 2m11s
307 [M21] math-medium verified 2m51s
308 [22] medium solved 2m03s
309 [24] medium solved 2m26s
310 [23] medium solved 3m16s
311 ▶ [30] hard solved 4m06s
312 [22] medium solved 2m08s
313 ▶ [29] hard solved 4m28s
314 ▶ [28] hard solved 1m44s
315 [20] medium solved 1m56s
316 [20] medium solved 5m28s
317 [22] medium solved 5m55s
318 ▶ [20] medium solved 2m19s
319 [21] medium solved 2m45s
320 ▶ [M38] math-project solved 5m
321 [42] project solved 1m38s
322 [25] medium solved 2m05s
323 [M25] math-medium solved 2m16s
324 ▶ [30] hard solved 3m14s
325 [27] hard solved 2m20s
326 ▶ [M25] math-medium solved 2m25s
327 [24] medium solved 2m29s
328 ▶ [M23] math-medium solved 4m45s
329 [22] medium solved 5m55s
330 [25] medium solved 4m28s
331 [M40] math-project solved 2m20s
332 [30] hard solved 4m39s
333 [21] medium solved 2m
334 ▶ [M32] math-hard solved 2m09s
335 [30] hard solved 3m18s
336 [21] medium solved 2m02s
337 [29] hard solved 3m07s
338 [22] medium solved 2m23s
339 [25] medium solved 5m58s
340 [30] hard solved 7m01s
341 ▶ [25] medium solved 3m51s
342 [25] medium solved 5m43s
343 [10] simple solved 9m11s
344 [10] simple solved 3m39s
345 [20] medium solved 3m56s
346 [M30] math-hard solved 5m22s
347 ▶ [M21] math-medium verified 1m41s
348 [M41] math-project solved 4m14s
349 ▶ [M27] math-hard verified 1m53s
350 [22] medium solved 6m08s
351 [M46] math-research solved 10m16s
352 [21] medium solved 4m30s
353 [39] project solved 3m31s
354 ▶ [M30] math-hard solved 5m03s
355 [25] medium solved 8m48s
356 [27] hard solved 5m41s
357 [M40] math-project solved 2m42s
358 [HM41] hm-project solved 34s
359 [29] hard solved 11m42s
360 ▶ [20] medium solved 2m05s
361 [M25] math-medium solved 6m24s
362 [10] simple solved 10m25s
363 [20] medium solved 1m49s
364 [23] medium solved 12m13s
365 [22] medium solved 7m53s
366 ▶ [25] medium solved 2m06s
367 [20] medium solved 2m27s
368 [M21] math-medium solved 3m07s
369 [27] hard solved 4m03s
370 ▶ [23] medium verified 11m18s
371 [24] medium solved 6m54s
372 ▶ [M35] math-hard solved 2m07s
373 [26] hard solved 1m04s
374 [M28] math-hard solved 3m24s
375 [M29] math-hard solved 3m22s
376 ▶ [M25] math-medium solved 5m48s
377 [M28] math-hard solved 6m17s
378 [M30] math-hard solved 3m21s
379 ▶ [25] medium solved 4m46s
380 [35] hard solved 5m08s
381 ▶ [20] medium verified 2m37s
382 [18] medium solved 1m23s
383 [29] hard solved 4m27s
384 [34] hard solved 3m10s
385 [M36] math-project solved 3m04s
386 ▶ [M31] math-hard solved 2m41s
387 ▶ [M26] math-hard solved 2m35s
388 ▶ [21] medium solved 1m50s
389 [29] hard solved 2m25s
390 ▶ [21] medium solved 2m12s
391 [29] hard solved 2m06s
392 ▶ [25] medium solved 2m40s
393 [25] medium solved 6m49s
394 [29] hard solved 7m18s
395 [25] medium solved 7m11s
396 ▶ [35] hard solved 3m02s
397 ▶ [30] hard solved 2m38s
398 [23] medium solved 2m07s
399 ▶ [22] medium solved 1m50s
400 [21] medium solved 2m01s
401 [22] medium solved 4m31s
402 [24] medium solved 2m19s
403 ▶ [31] hard solved 6m24s
404 ▶ [25] medium solved 1m51s
405 [21] medium verified 1m34s
406 [16] medium solved 4m53s
407 ▶ [20] medium solved 2m10s
408 [28] hard solved 6m58s
409 ▶ [30] hard solved 2m59s
410 [22] medium solved 5m15s
411 [20] medium solved 2m17s
412 ▶ [22] medium solved 2m
413 [30] hard verified 1m30s
414 [25] medium solved 3m13s
415 [M33] math-hard solved 5m11s
416 [M30] math-hard solved 2m31s
417 [M46] math-research solved 1m50s
418 [M29] math-hard solved 4m26s
419 [30] hard solved 5m26s
420 [M22] math-medium solved 2m42s
421 ▶ [20] medium solved 4m05s
422 [21] medium solved 2m15s
423 ▶ [M25] math-medium solved 1m58s
424 [36] project solved 1m59s
425 [25] medium solved 2m29s
426 ▶ [37] project solved 4m18s
427 ▶ [25] medium solved 2m21s
428 [M28] math-hard solved 2m55s
429 [21] medium solved 1m51s
430 ▶ [26] hard solved 2m44s
431 ▶ [30] hard solved 1m38s
432 ▶ [M25] math-medium solved 1m51s
433 [26] hard solved 1m40s
434 [39] project solved 3m25s
435 [27] hard solved 2m44s
436 ▶ [20] medium solved 2m24s
437 ▶ [27] hard solved 2m45s
438 [30] hard solved 2m32s
439 [M30] math-hard solved 2m14s
440 [21] medium solved 1m33s
441 [18] medium verified 1m43s
442 ▶ [M23] math-medium solved 3m19s
443 ▶ [M30] math-hard solved 4m37s
444 [M27] math-hard solved 5m12s
445 ▶ [M22] math-medium solved 1m49s
446 ▶ [44] project solved 1m43s
447 [22] medium solved 2m56s
448 [22] medium solved 4m59s
449 [40] project solved 2m11s
450 [42] project solved 2m16s
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 1

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 math-medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 2

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 math-hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 3

The system is interpreted exactly as written: x_2 + x_3 = x_3 + x_5 + x_6 = x_2 + x_5 = x_3 + x_4 = x_1 + x_4 = x_2 + x_3 + x_4 + x_6 = x_1 + x_6 = 1, with each $x_k \in {0,1}$ for $1 \le k \le 6$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 4

Let $G = (V, E)$ be a (simple, undirected) graph.
taocp mathematics algorithms volume-4 math-hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 5

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 6

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 simple Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 7

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 8

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 9

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 10

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 11

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 12

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 13

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 14

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 15

Let the positions be $1,2,\dots,2n$.
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TAOCP 7.2.2.1 Exercise 16

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 17

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 18

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 19

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 20

Let $m$ be the number of options in the pairwise ordering construction of (a6).
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TAOCP 7.2.2.1 Exercise 21

The flaw in the original solution is fundamental: it attempts to encode each index $j \in \{0,\dots,m-1\}$ using $L+1$ bits where $L=\lfloor \lg m \rfloor$, violating the requirement that each option...
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TAOCP 7.2.2.1 Exercise 22

An $n$-queens solution is a set $S \subseteq {1,\dots,n}^2$ with exactly one queen in each row and each column, satisfying the two diagonal constraints.
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TAOCP 7.2.2.1 Exercise 23

Let $n \times n$ chessboard coordinates be $(i,j)$ with $1 \le i,j \le n$.
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TAOCP 7.2.2.1 Exercise 24

An $n$-queens solution is a permutation $p$ of ${1,\dots,n}$ such that queens are placed at $(i,p(i))$ and no two attack each other.
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TAOCP 7.2.2.1 Exercise 25

Let $Q_8$ be the graph whose vertices are the $64$ squares of an $8\times 8$ chessboard, with two vertices adjacent when a queen placed on one square attacks the other along a row, column, or diagonal...
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TAOCP 7.2.2.1 Exercise 26

The original solution fails at the only place where the problem becomes genuinely global: it replaces a coupled partition problem by a product of independent 7-queen counts.
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TAOCP 7.2.2.1 Exercise 27

Let Langford’s problem be represented in the usual exact-cover form of Section 7.
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TAOCP 7.2.2.1 Exercise 28

Formula (27) expresses the estimated completion ratio in the form $\prod_{j=0}^{t} \frac{c_j}{t_j}$ with integers satisfying $1 \le c_j \le t_j$.
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TAOCP 7.2.2.1 Exercise 29

In particular, the missing points that must be fixed in a genuine solution are: 1.
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TAOCP 7.2.2.1 Exercise 30

All such trees can arise as backtrack trees of Algorithm X.
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TAOCP 7.2.2.1 Exercise 31

The two requested randomizations can be obtained by adding random choices before the deterministic parts of Algorithm X begin and by replacing the deterministic minimum selection in step X3 by a rando...
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TAOCP 7.2.2.1 Exercise 32

Edit **Solution.
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TAOCP 7.2.2.1 Exercise 33

Let the columns of $A$ correspond to the item set $U$, and let the rows of $A$ be the options of the original exact cover problem.
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TAOCP 7.2.2.1 Exercise 34

\textbf{Construction.
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TAOCP 7.2.2.1 Exercise 35

A mathematically correct solution cannot be written from the information provided because the exercise statement is incomplete.
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TAOCP 7.2.2.1 Exercise 36

Let $z_k=\operatorname{TOP}(x_k)$ denote the item chosen at level $k$ of Algorithm X.
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TAOCP 7.2.2.1 Exercise 37

Let $\langle g_n\rangle$ denote the lexicographically smallest solution to the $\infty$ queens problem.
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TAOCP 7.2.2.1 Exercise 38

Let $g_n$ denote the lexicographically smallest solution of the $\infty$ queens problem.
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TAOCP 7.2.2.1 Exercise 39

Let $m$ be the number of options and let $n$ be the number of items.
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TAOCP 7.2.2.1 Exercise 40

Edit Let the database after rows (1,\ldots,k-1) have been processed contain entries [ (s_j,c_j).
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TAOCP 7.2.2.1 Exercise 41

The corrected solution below replaces the unsupported pruning argument with the actual state-merging algorithm required by the exercise, gives an implementation outline, and describes a reproducible c...
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TAOCP 7.2.2.1 Exercise 42

The counting algorithm of exercise 40 loses information because each database entry $(s_j,c_j)$ stores only the number of ways to obtain the set $s_j$.
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TAOCP 7.2.2.1 Exercise 43

The exercise asks for formulas for the entries $a_{ij}$, $b_{ij}$, and $c_{ij}$ of the three sudoku squares in equation (28).
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TAOCP 7.2.2.1 Exercise 44

The first $33$ digits of $\pi$ are 3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0.
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TAOCP 7.2.2.1 Exercise 45

Let $S_t$ denote the exact-cover instance remaining after the first $t$ naked single moves have been performed.
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TAOCP 7.2.2.1 Exercise 46

The data needed to solve the exercise is missing.
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TAOCP 7.2.2.1 Exercise 47

A self-contained solution cannot be produced from the information given here because the actual candidate chart (32) is missing.
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TAOCP 7.2.2.1 Exercise 48

Edit Let (S) denote the candidate data represented in chart (33).
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TAOCP 7.2.2.1 Exercise 49

Parts (a) and (b) can be proved from the definitions given in the statement.
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TAOCP 7.2.2.1 Exercise 50

\textbf{Solution.
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TAOCP 7.2.2.1 Exercise 51

The argument based on splitting the two occurrences of $7$ into an $7$-class and an $8$-class is not valid in general.
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TAOCP 7.2.2.1 Exercise 52

Algorithm X does not have a single intrinsic running time on a sudoku instance.
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TAOCP 7.2.2.1 Exercise 53

Working
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TAOCP 7.2.2.1 Exercise 54

I cannot produce a complete solution for Exercise 7.
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TAOCP 7.2.2.1 Exercise 55

I cannot complete a rigorous solution of Exercise 7.
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TAOCP 7.2.2.1 Exercise 56

A sudoku puzzle is a $9\times9$ array whose entries are elements of ${1,\ldots,9}$, with the usual constraints that every row, every column, and every $3\times3$ box contains each digit exactly once.
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TAOCP 7.2.2.1 Exercise 57

A=\{1,2,3\},\qquad B=\{4,5,6\},\qquad C=\{7,8,9\}.
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TAOCP 7.2.2.1 Exercise 58

Working
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TAOCP 7.2.2.1 Exercise 59

Exercise 7.
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TAOCP 7.2.2.1 Exercise 60

Exercise 7.
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TAOCP 7.2.2.1 Exercise 61

The $5\times5$ gerechte design in (35a) has the regions \begin{array}{ccccc} 1&1&1&2&2\\ 1&1&5&2&2\\ 4&5&5&5&2\\
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TAOCP 7.2.2.1 Exercise 62

Solution to TAOCP 7.2.2.1 Exercise 62.
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TAOCP 7.2.2.1 Exercise 63

The statement supplied here is insufficient to determine the requested number.
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TAOCP 7.2.2.1 Exercise 64

I cannot give a mathematically reliable “complete worked solution” for Exercise 7.
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TAOCP 7.2.2.1 Exercise 65

The statement of Exercise 7.
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TAOCP 7.2.2.1 Exercise 66

The figures containing the two sets of nine cards are not available in the prompt.
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TAOCP 7.2.2.1 Exercise 67

Let the rows and columns of the $9\times9$ array be numbered $1,\ldots,9$.
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TAOCP 7.2.2.1 Exercise 68

Working
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TAOCP 7.2.2.1 Exercise 69

Exercise 7.
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TAOCP 7.2.2.1 Exercise 70

Let the upper left cell have coordinates $(1,1)$, with the first coordinate increasing downward and the second coordinate increasing to the right.
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TAOCP 7.2.2.1 Exercise 71

A 3-dimensional matching instance consists of three disjoint sets $X$, $Y$, and $Z$, together with a set $T\subseteq X\times Y\times Z$ of allowed triples.
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TAOCP 7.2.2.1 Exercise 72

Let $N(M)$ denote the number of complete Dominosa reconstructions of a matrix $M$.
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TAOCP 7.2.2.1 Exercise 73

A fully corrected solution with a numerical maximum cannot be produced from the supplied material, because the reviewer feedback assumes the existence of an extremal result but does not supply one.
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TAOCP 7.2.2.1 Exercise 74

I cannot produce a correct completed solution for this exercise from the information available.
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TAOCP 7.2.2.1 Exercise 75

Edit Write the operation temporarily by juxtaposition, so that (xy) denotes (x\circ y).
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TAOCP 7.2.2.1 Exercise 76

The exact cover formulation of exercise 75(d) already contains one option for each possible local consequence of the grope identity.
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TAOCP 7.2.2.1 Exercise 77

Let G=(V,E),\qquad H=(W,F), with
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TAOCP 7.2.2.1 Exercise 78

Solution to TAOCP 7.2.2.1 Exercise 78.
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TAOCP 7.2.2.1 Exercise 79

The statement of the exercise refers to equation (48), but equation (48) is not included in the supplied Section 7.
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TAOCP 7.2.2.1 Exercise 80

The statement of Exercise 7.
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TAOCP 7.2.2.1 Exercise 81

The statement is false.
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TAOCP 7.2.2.1 Exercise 82

The statement is **true**.
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TAOCP 7.2.2.1 Exercise 83

Let the first item selected by Algorithm C be the primary item $p$, and let its active options be ordered as $O_1,\ldots,O_k$.
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TAOCP 7.2.2.1 Exercise 84

Algorithm C can be modified by adding a bound on the largest option number that is permitted in a partial solution.
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TAOCP 7.2.2.1 Exercise 85

Let the options of the XCC problem be numbered $1,\ldots,M$.
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TAOCP 7.2.2.1 Exercise 86

The corrected solution is given below.
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TAOCP 7.2.2.1 Exercise 87

Let $W$ be the dictionary, consisting of words of length $n$.
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TAOCP 7.2.2.1 Exercise 88

Let $\text{WORDS}(W)$ denote the set of words whose rank in the frequency ordering is at most $W$.
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TAOCP 7.2.2.1 Exercise 89

A complete corrected solution cannot be obtained from the information in the proposed solution, because the proposed solution contains no data, and the exercise depends on a specific external dictiona...
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TAOCP 7.2.2.1 Exercise 90

The previous solution cannot be repaired by adding a few missing sentences, because its central claim of optimality depends on a computation that was never specified.
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TAOCP 7.2.2.1 Exercise 91

A Boolean function on variables $x_1,\ldots,x_4$ is represented by a $3$CNF formula exactly when its set of falsifying assignments is a union of subcubes of dimension at least $1$ in the $4$-dimension...
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TAOCP 7.2.2.1 Exercise 92

Solution to TAOCP 7.2.2.1 Exercise 92.
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TAOCP 7.2.2.1 Exercise 93

The exercise asks for the “best” five-letter examples, but the term “best” is not defined in the statement alone.
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TAOCP 7.2.2.1 Exercise 94

The required object is a binary cycle of length $16$, since the indices in the quadruples are taken modulo $16$.
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TAOCP 7.2.2.1 Exercise 95

Let S=\{y_1\ldots y_n: y_i\in\{0,1\},\ p\leq \nu(y_1\ldots y_n)\leq q\}.
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TAOCP 7.2.2.1 Exercise 96

\begin{array}{cccccccc} 0&0&0&0&1&0&1&1\\ 0&0&0&1&0&0&0&1\\ 1&0&0&0&1&0&1&1\\ 0&0&1&0&0&0&1&0\\
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TAOCP 7.2.2.1 Exercise 97

The supplied statement is still insufficient to determine the mathematical answer.
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TAOCP 7.2.2.1 Exercise 98

The reviewer feedback identifies the central issue correctly: the proposed chain construction cannot be repaired by merely changing the color assignments.
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TAOCP 7.2.2.1 Exercise 99

Edit The statement is false.
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TAOCP 7.2.2.1 Exercise 100

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 101

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 102

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 103

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 104

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 105

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 106

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 107

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 108

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 109

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 110

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 111

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 112

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 113

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 114

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 115

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 116

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 117

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 118

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 119

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 120

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 math-hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 121

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 math-hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 122

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 123

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 math-hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 124

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 125

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 math-medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 126

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 127

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 128

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 129

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 130

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 131

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 132

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 133

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 134

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 135

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 136

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 137

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 138

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 139

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 140

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 141

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 142

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 143

Stopped thinking
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TAOCP 7.2.2.1 Exercise 144

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 145

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 146

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 147

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 148

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Medium Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 149

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 150

We restart from a complete and explicit formulation.
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TAOCP 7.2.2.1 Exercise 151

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
Practice › Mathematics › TAOCP › TAOCP Vol 4: Combinatorial Algorithms › TAOCP 7.2.2.1: Dancing Links ›

TAOCP 7.2.2.1 Exercise 152

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
taocp mathematics algorithms volume-4 Hard Jul 5, 2026
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TAOCP 7.2.2.1 Exercise 153

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 154

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 155

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 156

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 157

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 158

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 159

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 160

Let a configuration be a placement of $n$ queens on an $n \times n$ board with one queen in each row and each column, satisfying the diagonal constraints.
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TAOCP 7.2.2.1 Exercise 161

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 162

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 163

In (12), the variable $p$ is used to traverse exactly the vertical list of nodes that correspond to active options containing item $i$.
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TAOCP 7.2.2.1 Exercise 164

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 165

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 166

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 167

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 168

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 169

Let the vertices of $G$ be $v_1, v_2, \dots, v_n$.
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TAOCP 7.2.2.1 Exercise 170

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 171

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 172

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 173

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 174

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 175

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 176

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 177

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 178

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 179

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 180

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 181

Assume \tilde{D}(5n+r)=4^n c_r-\frac{3}{4}, \qquad n\ge 2,\quad 0\le r<5.
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TAOCP 7.2.2.1 Exercise 182

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 183

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 184

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 185

A strict exact cover problem consists of options, each option containing exactly one primary item and any number of secondary items, such that every primary item is covered exactly once and each secon...
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TAOCP 7.2.2.1 Exercise 186

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 187

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 188

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 189

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 190

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 191

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 192

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 193

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 194

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 195

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 196

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 197

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 198

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 199

We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
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TAOCP 7.2.2.1 Exercise 200

We keep the algebraic setup but fix part (b) by replacing symbolic computation with a randomized polynomial identity test in a finite field.
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TAOCP 7.2.2.1 Exercise 201

Items are the vertices $X_1,\dots,X_n$ and $Y_1,\dots,Y_n$.
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TAOCP 7.2.2.1 Exercise 202

The statement of the exercise depends entirely on Figure 202, which is not present in the provided context.
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TAOCP 7.2.2.1 Exercise 203

Equation (95) defines $T \otimes T'$ as the binary operation that combines two search trees by grafting $T'$ onto the terminal structure of $T$, with identity element $\square$ (the single-node tree).
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TAOCP 7.2.2.1 Exercise 204

Let d=\deg(\alpha), \qquad d'=\deg(\alpha').
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TAOCP 7.2.2.1 Exercise 205

A fully corrected solution cannot be produced from the information provided, because the exercise statement is incomplete.
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TAOCP 7.2.2.1 Exercise 206

Let the dominance order on nodes be denoted by $\preceq$, and recall that a tree is **minimally dominant** if its root is minimal in this order among all nodes of the tree, i.
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TAOCP 7.2.2.1 Exercise 207

We correct the solution, focusing especially on part (c), which requires a precise interpretation of “exponential growth” in Algorithm X dynamics, and a justified role for both parameters.
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TAOCP 7.2.2.1 Exercise 208

Let the exact cover instance of Fig.
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TAOCP 7.2.2.1 Exercise 209

Let the instance of the exact cover problem consist of a set of items $I$, partitioned into two disjoint classes $I = U \cup V$, $U \cap V = \varnothing$, together with a family of options $\mathcal{O...
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TAOCP 7.2.2.1 Exercise 210

Let the three options be denoted $\alpha'$, $\beta'$, and $\gamma'$.
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TAOCP 7.2.2.1 Exercise 211

We analyze bipairs in the standard exact cover formulations of the Langford pair problem, the $n$ queens problem, and Sudoku.
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TAOCP 7.2.2.1 Exercise 212

Let primary items be linearly ordered.
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TAOCP 7.2.2.1 Exercise 213

Let the items be linearly ordered and let the restricted growth string of a partition be defined in the standard way: scanning items in increasing order, each item receives the index of the block in w...
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TAOCP 7.2.2.1 Exercise 214

Let a _string solution_ be a sequence of options produced by the search procedure, where the same underlying exact cover solution may appear in different orders depending on the choices made during ba...
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TAOCP 7.2.2.1 Exercise 215

Let $K_{2q+1}$ have vertex set $\{0,1,\dots,2q\}$.
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TAOCP 7.2.2.1 Exercise 216

In Exercise 215, the underlying instance is an exact cover formulation of a combinatorial structure on $K_{2q+1}$.
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TAOCP 7.2.2.1 Exercise 217

The previous solution failed because it never actually classifies bipairs; it only restates the problem in terms of abstract “delta sets” and then assumes the conclusions.
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TAOCP 7.2.2.1 Exercise 218

Understood.
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TAOCP 7.2.2.1 Exercise 219

Let $p$ and $q$ be primary items in an XCC instance.
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TAOCP 7.2.2.1 Exercise 220

Let $A$ be an exact cover problem in the sense of Section 7.
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TAOCP 7.2.2.1 Exercise 221

Let $S$ be the stack formed in step P7 after all options that begin with items already on the search stack have been examined.
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TAOCP 7.2.2.1 Exercise 222

Let item $i$ be the item to be deleted in step P7, and let $S$ denote the distinguished item whose occurrences determine which options are treated as exceptional in this stage.
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TAOCP 7.2.2.1 Exercise 223

Let $S$ denote the stack of options accumulated in step P7.
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TAOCP 7.2.2.1 Exercise 224

Let the items be $x_1, x_2, \dots, x_n$.
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TAOCP 7.2.2.1 Exercise 225

In Algorithm P, the number of options removed during a covering step equals the number of nodes eliminated from the vertical lists of items that are deleted together with the chosen item.
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TAOCP 7.2.2.1 Exercise 226

Let $a_1,\dots,a_{2n}$ be a Langford pairing, and define the reversed sequence by $a'_k = a_{2n+1-k}, \qquad 1 \le k \le 2n.$ For any function $f$, define $T_f = \sum_{k=1}^{2n} k\, f(a_k), \qquad T_f...
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TAOCP 7.2.2.1 Exercise 227

In the Langford pairing exact cover formulation for $n=4$, options are indexed lexicographically by $(k,i)$ where $k$ is the value and $i$ is the first position, with the second position $j=i+k+1$.
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TAOCP 7.2.2.1 Exercise 228

Let $a_1\ldots a_{2n}$ be a Langford pairing, so each symbol $j \in {1,\dots,n}$ appears exactly twice among the $a_k$, and if $a_k = a_{k'} = j$ with $k<k'$, then $k'-k=j+1$.
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TAOCP 7.2.2.1 Exercise 229

A Langford pairing of order $n$ is a sequence $a_1,\dots,a_{2n}$ containing each symbol $k \in {1,\dots,n}$ exactly twice, with the two occurrences separated by exactly $k$ positions, so that if the f...
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TAOCP 7.2.2.1 Exercise 230

Let each option $O$ in the instance of Fig.
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TAOCP 7.2.2.1 Exercise 231

Let $G$ denote the set of all cells in the grid.
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TAOCP 7.2.2.1 Exercise 232

Let a placement of 16 queens be an option set $S$ consisting of 16 chosen cells $(i,j)$, and let its cost under Algorithm $X^8$ be $w(S)=\sum_{(i,j)\in S} 8d(i,j).$ Since multiplication by the positiv...
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TAOCP 7.2.2.1 Exercise 233

Let the 16-queens problem of Fig.
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TAOCP 7.2.2.1 Exercise 234

Let the board be $n \times n$, and let the center be $\left(\frac{n+1}{2}, \frac{n+1}{2}\right).$ For a queen placed at $(i,j)$, the cost is $8d(i,j)^2,$ and in the standard geometric interpretation u...
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TAOCP 7.2.2.1 Exercise 235

Let the board be $16 \times 16$ with rows and columns indexed by $i,j \in {1,\dots,16}$.
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TAOCP 7.2.2.1 Exercise 236

Let the board be indexed by $1,\dots,n$ in both directions, and let the center be $c = (n+1)/2$.
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TAOCP 7.2.2.1 Exercise 237

Let a solution of the prime square problem be an $n \times n$ array $(x_{ij})$ of primes satisfying the defining constraints of the problem in the text, and let the product of the solution be $P = \pr...
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TAOCP 7.2.2.1 Exercise 238

Let the array entries be constrained by digit class as follows: each entry is either a 3-digit prime or an $n$-digit prime, and all entries are distinct.
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TAOCP 7.2.2.1 Exercise 239

A family ${S_1,\ldots,S_m}$ of subsets of ${1,\ldots,n}$ is given together with weights $(w_1,\ldots,w_m)$, where each $w_j>0$.
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TAOCP 7.2.2.1 Exercise 240

The original solution failed because it never used the actual USA-partition instance.
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TAOCP 7.2.2.1 Exercise 241

Algorithm $P^s$ is a specialization of a general backtracking scheme in which a partial solution is extended step by step and each extension is later undone before exploring alternative branches.
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TAOCP 7.2.2.1 Exercise 242

Let $G = (V,E)$ be the graph processed by the algorithm of exercise 7.
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TAOCP 7.2.2.1 Exercise 243

Let a solution consist of exactly $d$ options, and let the weight of option $k$ be $x_k$ for $1 \le k \le d$.
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TAOCP 7.2.2.1 Exercise 244

Let $G$ be an undirected graph on vertex set $V$.
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TAOCP 7.2.2.1 Exercise 245

Let $G$ be the USA graph on 48 states, and let $G'$ be the augmented graph obtained by adding vertex $\mathrm{DC}$ adjacent only to $\mathrm{MD}$ and $\mathrm{VA}$.
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TAOCP 7.2.2.1 Exercise 246

Let a partition consist of options $O_1,\dots,O_7$, each induced subgraph on its vertex set having size fixed by the construction in (118).
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TAOCP 7.2.2.1 Exercise 247

Let each option $O$ have original cost $c(O)\ge 0$.
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TAOCP 7.2.2.1 Exercise 248

Let $i$ be an active item, and let $f(i)$ denote the number of active options that contain $i$ and have cost strictly less than $\theta = T - C_l$ at the current level $l$ in step C3$^s$.
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TAOCP 7.2.2.1 Exercise 249

Let the costs be revealed as a sequence $x_1, x_2, \ldots, x_{dt}$, where $\{x_1,\ldots,x_{dt}\} = \{c_1,\ldots,c_{dt}\}$ and each $x_t \ge 0$.
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TAOCP 7.2.2.1 Exercise 250

Let $Z$ be a set of characters with the property that for each $\alpha \in Z$, every option contains exactly one primary item whose name begins with $\alpha$.
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TAOCP 7.2.2.1 Exercise 251

Algorithm Z operates by recursive search over partial exact covers, maintaining the invariant that the current data structure represents the residual exact cover instance induced by the choices alread...
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TAOCP 7.2.2.1 Exercise 252

Let (121) denote the set of options defining the exact cover instance, and let Algorithm Z construct a ZDD by recursive application of step Z3, where each node corresponds to a choice of an item $i$ a...
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TAOCP 7.2.2.1 Exercise 253

Let $Z$ denote Algorithm Z as in Section 7.
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TAOCP 7.2.2.1 Exercise 254

Let Algorithm Z operate on an exact cover instance with primary items and secondary items with colors, in the sense of Section 7.
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TAOCP 7.2.2.1 Exercise 255

Let $K_n$ denote the complete graph on vertex set ${1,2,\dots,n}$ and consider the exact cover formulation of perfect matchings where each item is a vertex and each option is an unordered pair ${i,j}$...
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TAOCP 7.2.2.1 Exercise 256

Algorithm Z reduces the problem of finding perfect matchings of a graph to an exact cover instance in which each vertex is an item and each edge is an option covering its two endpoints, with the addit...
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TAOCP 7.2.2.1 Exercise 257

The items are $1,2,\dots,n$.
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TAOCP 7.2.2.1 Exercise 258

The previous solution fails because it replaces Algorithm Z’s actual backtracking dynamics with a single-pass incidence count.
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TAOCP 7.2.2.1 Exercise 259

Each bounded permutation instance has items $X_1,\dots,X_n,Y_1,\dots,Y_n$ and options $O_{ij} = \{X_i, Y_j\} \qquad (1 \le j \le a_i).$ A solution is a set of options selecting exactly one $Y_j$ for e...
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TAOCP 7.2.2.1 Exercise 260

We address the reviewer’s objections by redoing the analysis from the structure of the two exact cover instances, and by separating clearly: 1.
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TAOCP 7.2.2.1 Exercise 261

Let $G=(V,E)$ be a directed acyclic graph, let $S \subseteq V$ be the set of sources and $T \subseteq V$ the set of sinks.
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TAOCP 7.2.2.1 Exercise 262

The shape $S_n$ is a $16 \times n$ rectangular region with four fixed right triangles of side $7$ removed from its corners.
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TAOCP 7.2.2.1 Exercise 263

Let $I$ be an exact-cover instance arising from a problem in which each solution is a set of rows covering all columns exactly once.
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TAOCP 7.2.2.1 Exercise 264

Let the items be arranged in the circular doubly linked list headed by node $0$, with the active items forming a linear order when read from $i = \mathrm{RLINK}(0)$ forward.
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TAOCP 7.2.2.1 Exercise 265

We consider Algorithm Z as described in Section 7.
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TAOCP 7.2.2.1 Exercise 266

The utility program reads a description of a target shape and a set of polyominoes, then outputs a list of *options* for an exact cover solver.
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TAOCP 7.2.2.1 Exercise 267

Let the Conway pentomino names be used in their standard letter forms $F, I, L, N, P, T, U, V, W, X, Y, Z$.
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TAOCP 7.2.2.1 Exercise 268

The problem is an exact cover instance in the sense of (6)–(9): each legal placement of a pentomino on the $5\times 12$ board corresponds to one option, and a valid tiling corresponds to a set of opti...
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TAOCP 7.2.2.1 Exercise 269

Let a decomposable packing be one in which a vertical line between columns $k$ and $k+1$ separates the $5\times 12$ rectangle into a $5\times k$ region and a $5\times(12-k)$ region, with no pentomino...
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TAOCP 7.2.2.1 Exercise 270

Let the 11 nonsquare pentominoes be the free pentomino set with the $O$ pentomino removed.
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TAOCP 7.2.2.1 Exercise 271

A pentomino tiling of a $6\times 10$ rectangle can be encoded as an exact cover problem in the sense of Algorithm X, with items representing both geometric constraints and piece constraints, and with...
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TAOCP 7.2.2.1 Exercise 272

In the exact cover formulation of pentomino packing, each option represents a placement of a specific pentomino, covering one item for the pentomino identity and five items for the occupied unit squar...
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TAOCP 7.2.2.1 Exercise 273

Let the $3\times 20$ board be fixed.
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TAOCP 7.2.2.1 Exercise 274

We restart from first principles and remove the two unsupported assumptions in the previous solution: 1.
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TAOCP 7.2.2.1 Exercise 275

Color the $8\times 8$ board in the standard checkerboard coloring and assign each square weight $+1$ for black and $-1$ for white.
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TAOCP 7.2.2.1 Exercise 276

Let the five tetrominoes be denoted by $I$ (straight), $O$ (square), $T$, $L$, and $S$ (skew).
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TAOCP 7.2.2.1 Exercise 277

We restate the problem in a form that separates what is purely structural from what must be verified finitely and explicitly.
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TAOCP 7.2.2.1 Exercise 278

Let $\mathcal{P}$ denote the set of all $6 \times 10$ pentomino packings obtained by Algorithm X without symmetry reduction.
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TAOCP 7.2.2.1 Exercise 279

Let the cube have edge length $\sqrt{10}$.
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TAOCP 7.2.2.1 Exercise 280

A Möbius strip of width $4$ formed from unit squares has fundamental domain a $4 \times 15$ rectangle, since each pentomino has area $5$ and the twelve pentominoes cover $60$ unit squares, so the tota...
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TAOCP 7.2.2.1 Exercise 281

The Aztec diamond of order $11/2$ contains $61$ cells, and the Aztec diamond of order $13/2$ with a hole of order $3/2$ contains $80$ cells.
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TAOCP 7.2.2.1 Exercise 282

The original argument fails because it replaces the geometric constraint system with an exact-cover abstraction and then draws global invariance conclusions that do not follow.
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TAOCP 7.2.2.1 Exercise 283

Let $P$ be a fixed pentomino.
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TAOCP 7.2.2.1 Exercise 284

Let $\mathcal{P}={I,L,P,N,T,U,V,W,X,Y,Z,O,F}$ be the twelve pentominoes, considered up to translation, rotation, and reflection.
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TAOCP 7.2.2.1 Exercise 285

Each one-sided pentomino is a connected 5-cell polyomino, and there are 18 distinct pieces.
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TAOCP 7.2.2.1 Exercise 286

Let the twelve pentominoes be the standard set, with each piece used exactly once to tile the $6\times 10$ rectangle.
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TAOCP 7.2.2.1 Exercise 287

Let each pentomino placement be an option $O$.
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TAOCP 7.2.2.1 Exercise 288

Each one-sided pentomino is a fixed 5-cell polyomino with orientation distinguished up to rotation, but not reflection.
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TAOCP 7.2.2.1 Exercise 289

Please provide Figure (36) and the full image for exercise 289(c), or the corresponding region coordinates.
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TAOCP 7.2.2.1 Exercise 290

Let the board be a rectangle whose cells are colored in the usual checkerboard fashion.
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TAOCP 7.2.2.1 Exercise 291

Solution to TAOCP 7.2.2.1 Exercise 291.
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TAOCP 7.2.2.1 Exercise 292

Color the infinite square grid as a checkerboard, assigning the two colors according to the parity of the coordinates of a cell.
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TAOCP 7.2.2.1 Exercise 293

Let a hexomino be represented by a finite connected set of six unit squares.
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TAOCP 7.2.2.1 Exercise 294

The missing information identified in the previous response remains a decisive obstacle.
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TAOCP 7.2.2.1 Exercise 295

The missing figure is essential data for this exercise.
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TAOCP 7.2.2.1 Exercise 296

Exercise 7.
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TAOCP 7.2.2.1 Exercise 297

Exercise 7.
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TAOCP 7.2.2.1 Exercise 298

There are $80$ cells in the $8\times10$ rectangle.
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TAOCP 7.2.2.1 Exercise 299

Let $R$ be the $5\times54$ rectangle.
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TAOCP 7.2.2.1 Exercise 300

The three parts have different logical status.
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TAOCP 7.2.2.1 Exercise 301

I’m not able to produce a reliable complete solution to this exercise without risking fabricated enumeration data.
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TAOCP 7.2.2.1 Exercise 302

Solution to TAOCP 7.2.2.1 Exercise 302.
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TAOCP 7.2.2.1 Exercise 303

A complete corrected solution would need, in addition to the generating-function derivation, one of the following for part (d): 1.
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TAOCP 7.2.2.1 Exercise 304

Let $\mathcal P$ denote the decision problem in the statement.
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TAOCP 7.2.2.1 Exercise 305

The numerical counts requested in exercise 305 cannot be derived from the information supplied here.
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TAOCP 7.2.2.1 Exercise 306

The exercise asks for an exact enumeration of arrangements of the ten windmill dominoes subject to two simultaneous snake-in-the-box cycle conditions.
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TAOCP 7.2.2.1 Exercise 307

Number the rows and columns of the rectangle starting with $0$.
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TAOCP 7.2.2.1 Exercise 308

A complete solution cannot be derived from the supplied section context alone.
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TAOCP 7.2.2.1 Exercise 309

The twelve hexiamonds have the following numbers of base placements.
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TAOCP 7.2.2.1 Exercise 310

Solution to TAOCP 7.2.2.1 Exercise 310.
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TAOCP 7.2.2.1 Exercise 311

In particular, a correct solution must contain all of the following concrete items: 1.
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TAOCP 7.2.2.1 Exercise 312

I cannot produce a correct solution to Exercise 7.
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TAOCP 7.2.2.1 Exercise 313

I cannot give a corrected numerical solution to this exercise without performing the actual enumeration.
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TAOCP 7.2.2.1 Exercise 314

Let the four pentiamonds be $P_1,P_2,P_3,P_4$.
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TAOCP 7.2.2.1 Exercise 315

Let the coordinates of the cells of a polyhex be given by the coordinate system of the infinite hexagonal grid in the exercise.
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TAOCP 7.2.2.1 Exercise 316

Analyzing
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TAOCP 7.2.2.1 Exercise 317

I cannot produce a mathematically valid corrected solution for this exercise from the information available here.
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TAOCP 7.2.2.1 Exercise 318

Use the coordinate system of Exercise 124 for the triangular grid.
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TAOCP 7.2.2.1 Exercise 319

T(x,y)=(x+y,x-y).
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TAOCP 7.2.2.1 Exercise 320

The corrected solution is given below in a textbook style, with the enumeration and verification steps made explicit.
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TAOCP 7.2.2.1 Exercise 321

A rigorous solution would have to: 1.
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TAOCP 7.2.2.1 Exercise 322

Exercise 265 extends Algorithm X to packing problems by making each possible placement of a piece into the container an option, with items representing the conditions that must be satisfied exactly on...
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TAOCP 7.2.2.1 Exercise 323

A skewed pixel diagram can be drawn by replacing the ordinary square grid with the checkerboard tiling formed by unit squares and unit rhombuses.
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TAOCP 7.2.2.1 Exercise 324

A base placement is a placement of a Soma piece in the $3\times3\times3$ cube.
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TAOCP 7.2.2.1 Exercise 325

Let $V$ be the set of $240$ equivalence classes of solutions of the Soma cube problem.
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TAOCP 7.2.2.1 Exercise 326

Assign coordinates $(x,y,z)$ to the cubies of Fig.
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TAOCP 7.2.2.1 Exercise 327

Solution to TAOCP 7.2.2.1 Exercise 327.
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TAOCP 7.2.2.1 Exercise 328

The statement of the exercise in the prompt contains a dimensional error.
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TAOCP 7.2.2.1 Exercise 329

Let the coordinates of the box be B=\{(x,y,z):1\le x\le 3,\ 1\le y\le 4,\ 1\le z\le 3\}.
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TAOCP 7.2.2.1 Exercise 330

A complete enumeration is most naturally done by reducing the question to a finite exact-cover computation.
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TAOCP 7.2.2.1 Exercise 331

Let a _Soma shape_ mean a connected set of $27$ unit cubes that can be tiled by the seven fixed Soma pieces, with congruent shapes identified under the symmetries of the cube.
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TAOCP 7.2.2.1 Exercise 332

I cannot produce a correct enumeration for this exercise from the information provided, because the defining figure for the three target shapes is not available in the conversation.
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TAOCP 7.2.2.1 Exercise 333

The previous solution had the right mechanical idea but treated the crucial verifications as if they were already done.
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TAOCP 7.2.2.1 Exercise 334

A complete solution to Exercise 7.
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TAOCP 7.2.2.1 Exercise 335

I cannot produce a mathematically valid corrected solution from the information supplied.
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TAOCP 7.2.2.1 Exercise 336

The statement supplied for exercise 336 is incomplete because the defining figure for the L-bert Hall piece is missing.
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TAOCP 7.2.2.1 Exercise 337

Use coordinates $(x,y,z)$ for the unit cubes of the large cube, where $0\le x,y,z<3$.
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TAOCP 7.2.2.1 Exercise 338

The statement refers to six target shapes shown in Figure 338, but the figure itself is not included in the supplied material.
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TAOCP 7.2.2.1 Exercise 339

Let $O$ be a free octomino, and let $P(O)$ be the $4$-level prism obtained by stacking four copies of $O$.
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TAOCP 7.2.2.1 Exercise 340

\textbf{Solution.
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TAOCP 7.2.2.1 Exercise 341

A complete solution to this exercise must exhibit actual packings.
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TAOCP 7.2.2.1 Exercise 342

Solution to TAOCP 7.2.2.1 Exercise 342.
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TAOCP 7.2.2.1 Exercise 343

Solution to TAOCP 7.2.2.1 Exercise 343.
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TAOCP 7.2.2.1 Exercise 344

\textbf{Solution.
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TAOCP 7.2.2.1 Exercise 345

The corrected solution is: Edit The supplied statement does not contain the defining data needed to determine the U-shaped dodecacube or the meaning of a forbidden cross.
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TAOCP 7.2.2.1 Exercise 346

A fully corrected solution cannot be produced reliably from the information available in the prompt alone.
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TAOCP 7.2.2.1 Exercise 347

Let the cells of the $l \times m \times n$ box have coordinates $(x,y,z)$, where $0\le x<l,\qquad 0\le y<m,\qquad 0\le z<n.$ Let $\omega$ be a primitive $k$th root of unity.
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TAOCP 7.2.2.1 Exercise 348

The reviewer’s principal objection is based on a misinterpretation of the exercise.
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TAOCP 7.2.2.1 Exercise 349

Let s=a+b+c, and consider the cube
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TAOCP 7.2.2.1 Exercise 350

The proposed slab argument is a valid reduction, but the rectangle packing used in the previous solution is not.
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TAOCP 7.2.2.1 Exercise 351

The proposed solution does not answer the stated exercise.
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TAOCP 7.2.2.1 Exercise 352

Each pentomino is regarded as a flat $5$-cell polycube embedded in the $2 \times 2 \times 3 \times 5$ hyperbox.
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TAOCP 7.2.2.1 Exercise 353

Corrected solution: Edit A weak polycube of size (3) is a connected set of three unit cubes whose centers are lattice points in (\mathbb Z^3).
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TAOCP 7.2.2.1 Exercise 354

I can write the requested rigorous solution, but the exercise is long and has several parts requiring derivations of specific matrices and proofs of the symmetry group statement.
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TAOCP 7.2.2.1 Exercise 355

Solution to TAOCP 7.2.2.1 Exercise 355.
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TAOCP 7.2.2.1 Exercise 356

Please provide the proposed solution and the reviewer feedback (paste the text or upload the files).
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TAOCP 7.2.2.1 Exercise 357

A truncated octahedron has $6$ square faces and $8$ hexagonal faces, so a polysplatt is determined by a connected set of cells in the truncated-octahedral honeycomb.
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TAOCP 7.2.2.1 Exercise 358

Represent the centers of the spheres by coordinates in the hexagonal stacking, using two-dimensional triangular coordinates inside each layer and a layer index.
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TAOCP 7.2.2.1 Exercise 359

The proposed solution does not answer the stated exercise.
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TAOCP 7.2.2.1 Exercise 360

Let the coordinates of the reduced $m \times n$ rectangle be 0,1,\ldots,m in the vertical direction and
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TAOCP 7.2.2.1 Exercise 361

Edit The minimum number of subrectangles in a reduced (m\times n) pattern is [ \boxed{m+n-1}.
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TAOCP 7.2.2.1 Exercise 362

A Boolean function on variables $x_1,\ldots,x_4$ is represented by a $3$CNF formula exactly when its set of falsifying assignments is a union of subcubes of dimension at least $1$ in the $4$-dimension...
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TAOCP 7.2.2.1 Exercise 363

A decomposition of an $m \times n$ rectangle into grid-aligned subrectangles can be represented as an exact cover problem.
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TAOCP 7.2.2.1 Exercise 364

**True.
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TAOCP 7.2.2.1 Exercise 365

**Primary items.
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TAOCP 7.2.2.1 Exercise 366

Edit Let the construction of Exercise 363 be regarded as a rooted search tree.
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TAOCP 7.2.2.1 Exercise 367

Let a motley dissection of an $m\times n$ rectangle be represented by the closed coordinate intervals of its subrectangles.
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TAOCP 7.2.2.1 Exercise 368

Let the $m\times n$ rectangle be divided into $t$ subrectangles.
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TAOCP 7.2.2.1 Exercise 369

The data supplied do not contain enough information to produce a valid complete solution with the numerical maxima.
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TAOCP 7.2.2.1 Exercise 370

Please provide the proposed solution and the reviewer feedback (paste the text or upload the files).
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TAOCP 7.2.2.1 Exercise 371

R=[a\ldots b)\times[c\ldots d) denotes a rectangle whose horizontal interval is $[a\ldots b)$ and whose vertical interval is $[c\ldots d)$.
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TAOCP 7.2.2.1 Exercise 372

Edit Let (r \ge r') denote reachability through a chain of horizontal walls, with each step going from a room to the room immediately below it.
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TAOCP 7.2.2.1 Exercise 373

Understood.
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TAOCP 7.2.2.1 Exercise 374

Edit Let the rectangles of an incomparable dissection be (R_i), with dimensions (h_i\times w_i).
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TAOCP 7.2.2.1 Exercise 375

A complete corrected solution cannot be written from the information supplied in the prompt.
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TAOCP 7.2.2.1 Exercise 376

\textbf{Solution.
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TAOCP 7.2.2.1 Exercise 377

A rectangle $h\times w$ will always mean a rectangle with positive integer side lengths.
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TAOCP 7.2.2.1 Exercise 378

Edit Let a rectangular shape be denoted by $h\times w$, where $h,w\in\mathbb N$.
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TAOCP 7.2.2.1 Exercise 379

The empty submission gives no information, so the solution must begin by determining the finite basis of packable rectangles for the $Q$-pentomino.
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TAOCP 7.2.2.1 Exercise 380

Edit Let (Y) denote the pentomino consisting of a column of four cells with one additional cell attached to the second cell of the column.
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TAOCP 7.2.2.1 Exercise 381

Place coordinates on the $12 \times n$ rectangle, with rows numbered $1,2,\ldots,12$ and columns numbered $1,2,\ldots,n$.
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TAOCP 7.2.2.1 Exercise 382

The construction cannot be recovered from the information supplied in the exercise statement alone.
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TAOCP 7.2.2.1 Exercise 383

A complete solution to Exercise 7.
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TAOCP 7.2.2.1 Exercise 384

The corrected solution must include both the exact-cover construction and the actual enumeration for the case $l=m=n=7$.
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TAOCP 7.2.2.1 Exercise 385

The statement is not presently proved.
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TAOCP 7.2.2.1 Exercise 386

A symmetry of a polyiamond or a polyhex is an element of the symmetry group of the triangular lattice or hexagonal lattice.
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TAOCP 7.2.2.1 Exercise 387

A polycube has a symmetry group consisting of those rotations of space that preserve the set of cubes.
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TAOCP 7.2.2.1 Exercise 388

The three futoshiki instances in Figure 388 are required in order to produce the worked solutions.
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TAOCP 7.2.2.1 Exercise 389

Let the entries of an $n\times n$ futoshiki puzzle be denoted by $x_{r,c}$, with every entry satisfying $1\le x_{r,c}\le n.$ Each row and column contains each of the values $1,\ldots,n$ exactly once.
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TAOCP 7.2.2.1 Exercise 390

Edit Let the entries of an (n\times n) futoshiki puzzle be (x_{r,c}), where [ 1\le r,c\le n,\qquad x_{r,c}\in{1,\ldots,n}.
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TAOCP 7.2.2.1 Exercise 391

The corrected solution is given below.
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TAOCP 7.2.2.1 Exercise 392

I cannot produce a mathematically valid corrected solution with the requested numerical table and examples from the information available here.
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TAOCP 7.2.2.1 Exercise 393

A complete correction requires an exhaustive enumeration.
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TAOCP 7.2.2.1 Exercise 394

Working
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TAOCP 7.2.2.1 Exercise 395

Consider the Latin square L= \begin{pmatrix} 1&3&2&5&4\\ 4&1&3&2&5\\
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TAOCP 7.2.2.1 Exercise 396

A $9\times9$ futoshiki solution is a Latin square on the symbols ${1,2,\ldots,9}$, together with the required strong and weak clues.
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TAOCP 7.2.2.1 Exercise 397

Let the grid cells be indexed by $(r,c)$, where $1\le r,c\le n$.
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TAOCP 7.2.2.1 Exercise 398

I can write the complete solution, but the data needed to solve it is missing: Figure 398, which defines the three KenKen puzzles (a), (b), and (c), is not included in the prompt.
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TAOCP 7.2.2.1 Exercise 399

Algorithm C can be applied after converting the KenKen puzzle into an exact cover problem.
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TAOCP 7.2.2.1 Exercise 400

The statement of Exercise 7.
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TAOCP 7.2.2.1 Exercise 401

Working
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TAOCP 7.2.2.1 Exercise 402

The exercise refers to a $12\times12$ KenKen puzzle whose cage layout is given in a figure.
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TAOCP 7.2.2.1 Exercise 403

Working
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TAOCP 7.2.2.1 Exercise 404

A hidato solution is a Hamiltonian path of king moves on the $m \times n$ board.
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TAOCP 7.2.2.1 Exercise 405

Let the graph be $G=(V,E)$, and let $v\in V$ be the specified starting vertex.
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TAOCP 7.2.2.1 Exercise 406

The first step is to notice that the statement as printed cannot be correct.
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TAOCP 7.2.2.1 Exercise 407

The statement of the exercise as provided is incomplete.
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TAOCP 7.2.2.1 Exercise 408

Working
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TAOCP 7.2.2.1 Exercise 409

We interpret the first $20$ digits of $\pi$ as ten two-digit clue values.
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TAOCP 7.2.2.1 Exercise 410

Let the $5\times5$ diagram mean the usual slitherlink board with $5\times5$ vertices, hence $4\times4$ cells.
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TAOCP 7.2.2.1 Exercise 411

Edit The statement is false.
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TAOCP 7.2.2.1 Exercise 412

Edit Use the coordinate convention suggested by the hint.
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TAOCP 7.2.2.1 Exercise 413

In the construction of exercise 412, the vertices of the slitherlink grid are represented by items that enforce the local degree condition.
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TAOCP 7.2.2.1 Exercise 414

The flaw in the previous argument was that it treated the missing diagram as an obstacle instead of analyzing the counterexample.
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TAOCP 7.2.2.1 Exercise 415

I cannot produce a correct completed solution to this exercise from the information available here, because the required numerical enumeration results are not contained in the exercise statement or re...
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TAOCP 7.2.2.1 Exercise 416

A complete answer would need, at minimum: 1.
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TAOCP 7.2.2.1 Exercise 417

Exercise 7.
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TAOCP 7.2.2.1 Exercise 418

I cannot produce a correct completed solution for parts (b)–(e) without carrying out the required exhaustive enumeration or having the enumeration output.
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TAOCP 7.2.2.1 Exercise 419

The displayed array is not merely a matter of omitted blank cells.
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TAOCP 7.2.2.1 Exercise 420

Let the cells be indexed by $(i,j)$, with $0\le i<m$ and $0\le j<n$.
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TAOCP 7.2.2.1 Exercise 421

Denote a cell by its two coordinates, as in the statement.
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TAOCP 7.2.2.1 Exercise 422

Let the cells of the Masyu puzzle be the vertices of the graph $G$ whose edges join orthogonally adjacent cells.
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TAOCP 7.2.2.1 Exercise 423

The construction in exercise 422 uses one Boolean variable $x_e$ for every potential edge $e$.
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TAOCP 7.2.2.1 Exercise 424

Let the cells of the $6 \times 6$ board be denoted by C=\{(i,j):0\leq i,j<6\}.
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TAOCP 7.2.2.1 Exercise 425

The supplied section gives the general Dancing Links machinery, but it does not contain the definitions of the five solution-tile symbols, the example $3\times3$ solution diagram, or the precise graph...
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TAOCP 7.2.2.1 Exercise 426

The exercise refers to a concrete diagram, namely diagram (i), whose initial arrangement of white clues must be modified.
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TAOCP 7.2.2.1 Exercise 427

I cannot produce a correct worked solution for Exercise 7.
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TAOCP 7.2.2.1 Exercise 428

A Masyu loop is a closed curve through cell centers.
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TAOCP 7.2.2.1 Exercise 429

The statement of Exercise 7.
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TAOCP 7.2.2.1 Exercise 430

The supplied statement does not include the two diagrams referred to in parts (a) and (c).
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TAOCP 7.2.2.1 Exercise 431

The statement of Exercise 7.
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TAOCP 7.2.2.1 Exercise 432

The numerical answer depends on the two diagrams in Figure 432.
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TAOCP 7.2.2.1 Exercise 433

The figure containing the kakuro grid is not present in the supplied material, so the numerical enumeration cannot be carried out without inventing missing data.
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TAOCP 7.2.2.1 Exercise 434

The black top row and left column are fixed.
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TAOCP 7.2.2.1 Exercise 435

A kakuro block is a maximal horizontal or vertical run of white cells.
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TAOCP 7.2.2.1 Exercise 436

Let a cell be **white** when it is not crossed out and **black** when it is crossed out.
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TAOCP 7.2.2.1 Exercise 437

Let the cells of the hitori array be denoted by $x=(r,c)$.
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TAOCP 7.2.2.1 Exercise 438

The corrected solution removes the invalid pruning argument and uses only a connectivity test that is guaranteed to be valid for partial assignments.
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TAOCP 7.2.2.1 Exercise 439

Let $G=(V,E)$ be a graph, and let $U\subseteq V$ satisfy the three conditions in the definition of a hitori cover.
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TAOCP 7.2.2.1 Exercise 440

The statement is false.
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TAOCP 7.2.2.1 Exercise 441

Let the $1\times n$ puzzle be the string a_1a_2\cdots a_n, where each $a_i$ is one of the $d$ letters in the alphabet.
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TAOCP 7.2.2.1 Exercise 442

Working
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TAOCP 7.2.2.1 Exercise 443

Let $B$ be the set of black cells and $W$ the set of white cells.
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TAOCP 7.2.2.1 Exercise 444

Let $B$ be the set of black cells in a valid $n\times n$ hitori cover, and let $W$ be the set of white cells.
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TAOCP 7.2.2.1 Exercise 445

You've hit your limit.
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TAOCP 7.2.2.1 Exercise 446

The exercise, as stated, asks for the probability that a random $5\times5$ array of digits defines a valid hitori puzzle, that is, has exactly one solution.
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TAOCP 7.2.2.1 Exercise 447

\boxed{m=2,\qquad n=2} is already enough.
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TAOCP 7.2.2.1 Exercise 448

\textbf{Answer.
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TAOCP 7.2.2.1 Exercise 449

A hitori puzzle is a rectangular array of symbols together with the rule that some cells are marked black so that no two black cells share an edge, and the remaining white cells contain no repeated sy...
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TAOCP 7.2.2.1 Exercise 450

The problem of Section 7.
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