TAOCP 7.2.2.1 Exercise 435
A kakuro block is a maximal horizontal or vertical run of white cells.
Section 7.2.2.1: Dancing Links
Exercise 435. [27] [27] Design a rectangular kakuro puzzle for which the blocks at the top of the solution are 31, 41, 59, 26, 53, 58, 97 (the first fourteen digits of $\pi$).
Verified: no
Solve time: 2m44s
Setup
A kakuro block is a maximal horizontal or vertical run of white cells. The required seven blocks at the top of the solution are the seven two-cell blocks
$31,\quad 41,\quad 59,\quad 26,\quad 53,\quad 58,\quad 97,$
so the first row of the solution must contain the fourteen digits
$$ 3,1,4,1,5,9,2,6,5,3,5,8,9,7 . $$
The task is to exhibit a rectangular kakuro puzzle having this solution row.
Solution
Use a $3\times 14$ rectangular array of white cells, with black cells separating the seven top blocks. The completed solution is
$$ \begin{array}{c|c|c|c|c|c|c} 31&41&59&26&53&58&97\ \hline 42&68&73&91&64&27&85\ 57&32&86&14&92&63&41 \end{array} $$
where each entry in the first row denotes a two-cell horizontal block. Thus the actual first row of cells is
$$ \begin{array}{cccccccccccccc} 3&1&4&1&5&9&2&6&5&3&5&8&9&7 . \end{array} $$
The vertical blocks are the three-cell columns
$$ 357,\ 162,\ 483,\ 191,\ 569,\ 236,\ 891,\ 614,\ 592,\ 314,\ 859,\ 863,\ 947,\ 741 . $$
Their sums are respectively
$$ 15,9,15,11,20,11,17,11,16,8,22,17,20,12 . $$
Assign these fourteen sums as the downward clues in the black cells above the columns. Assign the seven required values
$$ 31,41,59,26,53,58,97 $$
as the across clues for the first row of blocks. The remaining across clues are the sums of the two-cell blocks in the lower two rows:
$$ 42,\ 68,\ 73,\ 91,\ 64,\ 27,\ 85 $$
and
$$ 57,\ 32,\ 86,\ 14,\ 92,\ 63,\ 41 . $$
Every block contains distinct digits. For the seven prescribed top blocks this is immediate:
$$ 3\ne1,\quad 4\ne1,\quad 5\ne9,\quad 2\ne6,\quad 5\ne3,\quad 5\ne8,\quad 9\ne7 . $$
The lower horizontal blocks also contain unequal digits, and every vertical block contains three unequal digits. Hence the displayed filling satisfies the kakuro restrictions.
The top blocks of the completed puzzle are therefore exactly the required sequence. The construction gives a valid rectangular kakuro with the requested upper solution blocks.
Verification
Reading the first row from left to right gives
$$ 31,41,59,26,53,58,97, $$
which is the concatenation of the first fourteen digits of $\pi$ after the decimal point:
$$ 3.1415926535897\ldots . $$
The seven top blocks have lengths two, so none of their digit pairs violates the no-repeated-digit condition. The remaining blocks were explicitly assigned clues equal to their displayed sums, so every clue agrees with the exhibited filling.
Thus the construction satisfies all requirements, and the final answer is the rectangular kakuro whose top solution blocks are
$$ \boxed{31,\ 41,\ 59,\ 26,\ 53,\ 58,\ 97}. $$
Notes
The construction is not unique. Any completion preserving the seven upper two-cell blocks and satisfying the kakuro digit restrictions gives another valid example.