Ken's Puzzle of the Week
Talisman Squares
Place the numbers 1-16 into a 4x4 grid such that the difference between a
number and any of its neighbors (including diagonally) is not less than 3.
In how many ways can this be done? (Disregard rotations, reflections, and
replacement with 17-x.)
Source: Original, based on several sources (formerly titled "Largest Difference Square".)
Solutions were received from Kirk Bresniker (who reported 173726), Mark
Rickert (who reported 112356), and Dan Chirica (who reported "many".)
Mark was able to restrict the problem by requiring the maximum difference in
the square to be as small as possible (results below.)Some references to
Talisman squares:
http://mathworld.wolfram.com/TalismanSquare.html
http://primepuzzles.net/puzzles/puzz_267.htm
Mark's solutions:
Assuming I've coded this properly, if you limit the differences to being
between 3 and 12 (inclusive), there are 26 solutions. Here's what I
got:
1 10 3 11
13 6 15 7
2 9 12 4
14 5 16 8
----------
1 10 3 11
13 6 15 8
2 9 12 4
14 5 16 7
----------
1 13 6 9
5 10 3 15
2 14 7 12
8 11 4 16
----------
1 13 9 12
10 6 3 15
2 14 11 7
5 8 4 16
----------
1 4 16 5
13 9 12 8
2 6 3 15
10 14 11 7
----------
1 4 16 5
13 9 12 8
2 6 3 15
11 14 10 7
----------
1 13 4 16
5 10 7 12
2 14 3 15
9 6 11 8
----------
1 13 4 16
5 10 7 12
2 14 3 15
8 11 6 9
----------
1 5 16 4
13 9 12 7
2 6 3 15
10 14 11 8
----------
1 5 16 4
13 9 12 8
2 6 3 15
10 14 11 7
----------
1 5 16 4
13 9 12 8
2 6 3 15
11 14 10 7
----------
1 5 16 7
13 9 12 4
2 6 15 8
14 10 3 11
----------
1 5 16 8
13 9 12 4
2 6 15 7
14 10 3 11
----------
1 5 16 8
13 9 12 4
2 6 15 7
14 11 3 10
----------
1 9 16 7
13 5 12 4
10 2 8 15
6 14 11 3
----------
5 9 6 3
13 1 12 15
10 7 4 8
2 14 11 16
----------
10 6 9 3
13 1 12 15
5 8 4 7
2 14 11 16
----------
5 8 12 9
13 1 4 16
6 10 7 11
2 14 3 15
----------
5 9 12 8
13 1 4 16
6 10 7 11
2 14 3 15
----------
5 9 12 16
13 1 4 8
10 7 11 15
2 14 3 6
----------
6 9 12 8
13 1 4 16
5 10 7 11
2 14 3 15
----------
10 7 11 16
13 1 4 8
5 9 12 15
14 2 6 3
----------
10 13 7 16
6 1 4 11
9 12 8 14
3 15 5 2
----------
13 10 7 16
6 1 4 11
9 12 8 14
3 15 5 2
----------
5 8 12 9
13 1 4 16
6 10 7 11
15 3 14 2
----------
5 9 12 8
13 1 4 16
6 10 7 11
15 3 14 2
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