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.
Section 7.2.2.1: Dancing Links
Exercise 418. [M29] [M29] An $m \times n$ slitherlink diagram is said to be symmetrical if cells $(i, j)$ and $(m - 1 - i, n - 1 - j)$ are both blank or both nonblank, for $0 \le i < m$ and $0 \le j < n$. (Many grid-based puzzles obey this oft-unwritten rule.)
a) There are exactly $6^{25} \approx 2.8 \times 10^{19}$ slitherlink diagrams of size $5 \times 5$, since each of the 25 cells can contain either '0', '1', '2', '3', '4', or ' '. How many of those diagrams are symmetrical?
b) How many of the symmetrical diagrams in (a) are valid puzzles?
c) How many of those valid puzzles are minimal, in the sense that the deletion of nonblank clues in $(i, j)$ and $(1, 4 - j)$ would make the solution nonunique?
d) What is the minimum number of clues in a valid $5 \times 5$ symmetrical puzzle?
e) What is the maximum number of clues in a minimal $5 \times 5$ symmetrical puzzle?
Verified: no
Solve time: 4m26s
I cannot produce a correct completed solution for parts (b)–(e) without carrying out the required exhaustive enumeration or having the enumeration output. The previous submission contains no data from which those values can be recovered.
The part (a) calculation can be completed directly.
A $5\times5$ diagram has $25$ cells. Under the symmetry map
$$ (i,j)\longmapsto (4-i,4-j), $$
there is one fixed cell, namely $(2,2)$, and the remaining $24$ cells form $12$ pairs.
The centre cell may contain any of the six symbols
$$ 0,1,2,3,4,\text{ or blank}, $$
so it contributes a factor of $6$.
For each of the twelve paired positions, the two cells must either both be blank or both be nonblank. If they are blank there is one possibility. If they are nonblank, each cell may independently contain one of the five digits $0,1,2,3,4$, giving
$$ 5^2=25 $$
possibilities. Thus each pair contributes
$$ 1+25=26 $$
possibilities.
Therefore the number of symmetrical diagrams is
$$ 6\cdot 26^{12}. $$
Since
$$ 26^{12}=95428956661682176, $$
we obtain
$$ \boxed{6\cdot26^{12}=572573739970093056}. $$
For parts (b)–(e), the missing work is an exhaustive search over the remaining symmetric clue patterns. A valid solution must enumerate all symmetric clue assignments, compute all slitherlink loops satisfying the clues, test uniqueness, and then test minimality by deleting each symmetric pair of clues. No purely local argument supplies the requested counts, and the previous answer gives no enumeration results. A complete solution therefore requires the actual search output or an independently reproducible enumeration.