Puzzled by a puzzle
- To: mathgroup at smc.vnet.net
- Subject: [mg113485] Puzzled by a puzzle
- From: Fred Simons <f.h.simons at tue.nl>
- Date: Sat, 30 Oct 2010 04:39:39 -0400 (EDT)
Consider the well known pentomino puzzle on the chess board without the four corners. For a reference see http://isomerdesign.com/Pentomino/, puzzle 8x8d. Apart from rotations and reflections, this puzzle has 2170 solutions, of which only 7 have the I-pentomino with one endpoint at one of the sides at position 3. The first time I wrote a function for solving this puzzle with backtracking must be about 20 years ago, and the last time I did it with Mathematica was in 2003. The program I used that time (in Mathematica 3 or 4) still runs under Mathematica 7. With the aim of compiling, I rewrote the function, and found all 2170 solutions in a little bit more than one and a half hour and the 7 solutions with the I at a side at position 3 in 30 seconds. So far so good. Then I compiled the function. The resulting function turned out to be about six times as faster than the identical uncompiled one. Obviously, I expected that the results would be the same. But for the I-pentomino the compiled function found in 5 seconds 15 solutions instead of 7. So my first impression was that there is a bug in compiling and that 8 of the 15 found solutions would be incorrect. But when I inspected the result, all 15 solutions seemed to be correct. If you want to see them, evaluate the following command. The solutions are in lexicographical order. {Graphics[{{Polygon[{{1, 7}, {1, 8}, {0, 8}, {0, 7}}], Polygon[{{8, 7}, {8, 8}, {7, 8}, {7, 7}}], Polygon[{{1, 5}, {1, 6}, {0, 6}, {0, 5}}], Polygon[{{2, 5}, {2, 6}, {1, 6}, {1, 5}}], Polygon[{{3, 5}, {3, 6}, {2, 6}, {2, 5}}], Polygon[{{4, 5}, {4, 6}, {3, 6}, {3, 5}}], Polygon[{{5, 5}, {5, 6}, {4, 6}, {4, 5}}], Polygon[{{1, 0}, {1, 1}, {0, 1}, {0, 0}}], Polygon[{{8, 0}, {8, 1}, {7, 1}, {7, 0}}]}, Line[{{{0, 0}, {0, 8}, {8, 8}, {8, 0}, {0, 0}}, {{1, 8}, {2, 8}}, {{1, 8}, {1, 7}}, {{2, 8}, {3, 8}}, {{3, 8}, {4, 8}}, {{4, 8}, {4, 7}}, {{0, 7}, {1, 7}}, {{0, 7}, {0, 6}}, {{2, 7}, {3, 7}}, {{2, 7}, {2, 6}}, {{3, 7}, {4, 7}}, {{0, 6}, {1, 6}}, {{1, 6}, {2, 6}}, {{4, 8}, {5, 8}}, {{4, 8}, {4, 7}}, {{5, 8}, {5, 7}}, {{2, 7}, {3, 7}}, {{2, 7}, {2, 6}}, {{3, 7}, {4, 7}}, {{5, 7}, {6, 7}}, {{6, 7}, {6, 6}}, {{2, 6}, {3, 6}}, {{3, 6}, {4, 6}}, {{4, 6}, {5, 6}}, {{5, 6}, {6, 6}}, {{5, 8}, {6, 8}}, {{5, 8}, {5, 7}}, {{6, 8}, {7, 8}}, {{7, 8}, {7, 7}}, {{5, 7}, {6, 7}}, {{6, 7}, {6, 6}}, {{7, 7}, {8, 7}}, {{8, 7}, {8, 6}}, {{6, 6}, {7, 6}}, {{7, 6}, {7, 5}}, {{8, 6}, {8, 5}}, {{7, 5}, {8, 5}}, {{5, 6}, {6, 6}}, {{5, 6}, {5, 5}}, {{6, 6}, {7, 6}}, {{7, 6}, {7, 5}}, {{4, 5}, {5, 5}}, {{4, 5}, {4, 4}}, {{7, 5}, {7, 4}}, {{4, 4}, {5, 4}}, {{5, 4}, {6, 4}}, {{6, 4}, {7, 4}}, {{0, 5}, {1, 5}}, {{0, 5}, {0, 4}}, {{1, 5}, {2, 5}}, {{2, 5}, {2, 4}}, {{0, 4}, {0, 3}}, {{1, 4}, {2, 4}}, {{1, 4}, {1, 3}}, {{0, 3}, {0, 2}}, {{1, 3}, {1, 2}}, {{0, 2}, {0, 1}}, {{1, 2}, {1, 1}}, {{0, 1}, {1, 1}}, {{2, 5}, {3, 5}}, {{2, 5}, {2, 4}}, {{3, 5}, {4, 5}}, {{4, 5}, {4, 4}}, {{1, 4}, {2, 4}}, {{1, 4}, {1, 3}}, {{3, 4}, {4, 4}}, {{3, 4}, {3, 3}}, {{1, 3}, {2, 3}}, {{2, 3}, {2, 2}}, {{3, 3}, {3, 2}}, {{2, 2}, {3, 2}}, {{7, 5}, {8, 5}}, {{7, 5}, {7, 4}}, {{8, 5}, {8, 4}}, {{5, 4}, {6, 4}}, {{5, 4}, {5, 3}}, {{6, 4}, {7, 4}}, {{8, 4}, {8, 3}}, {{5, 3}, {6, 3}}, {{6, 3}, {7, 3}}, {{7, 3}, {7, 2}}, {{8, 3}, {8, 2}}, {{7, 2}, {8, 2}}, {{3, 4}, {4, 4}}, {{3, 4}, {3, 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Polygon[{{2, 5}, {2, 6}, {1, 6}, {1, 5}}], Polygon[{{3, 5}, {3, 6}, {2, 6}, {2, 5}}], Polygon[{{4, 5}, {4, 6}, {3, 6}, {3, 5}}], Polygon[{{5, 5}, {5, 6}, {4, 6}, {4, 5}}], Polygon[{{1, 0}, {1, 1}, {0, 1}, {0, 0}}], Polygon[{{8, 0}, {8, 1}, {7, 1}, {7, 0}}]}, Line[{{{0, 0}, {0, 8}, {8, 8}, {8, 0}, {0, 0}}, {{1, 8}, {2, 8}}, {{1, 8}, {1, 7}}, {{2, 8}, {3, 8}}, {{3, 8}, {4, 8}}, {{4, 8}, {4, 7}}, {{0, 7}, {1, 7}}, {{0, 7}, {0, 6}}, {{2, 7}, {3, 7}}, {{2, 7}, {2, 6}}, {{3, 7}, {4, 7}}, {{0, 6}, {1, 6}}, {{1, 6}, {2, 6}}, {{4, 8}, {5, 8}}, {{4, 8}, {4, 7}}, {{5, 8}, {5, 7}}, {{2, 7}, {3, 7}}, {{2, 7}, {2, 6}}, {{3, 7}, {4, 7}}, {{5, 7}, {6, 7}}, {{6, 7}, {6, 6}}, {{2, 6}, {3, 6}}, {{3, 6}, {4, 6}}, {{4, 6}, {5, 6}}, {{5, 6}, {6, 6}}, {{5, 8}, {6, 8}}, {{5, 8}, {5, 7}}, {{6, 8}, {7, 8}}, {{7, 8}, {7, 7}}, {{5, 7}, {6, 7}}, {{6, 7}, {6, 6}}, {{7, 7}, {8, 7}}, {{8, 7}, {8, 6}}, {{6, 6}, {7, 6}}, {{7, 6}, {7, 5}}, {{8, 6}, {8, 5}}, {{7, 5}, {8, 5}}, {{5, 6}, {6, 6}}, {{5, 6}, {5, 5}}, {{6, 6}, 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{{2, 1}, {3, 1}}, {{3, 1}, {4, 1}}, {{4, 1}, {4, 0}}, {{5, 1}, {5, 0}}, {{4, 0}, {5, 0}}, {{5, 3}, {6, 3}}, {{5, 3}, {5, 2}}, {{6, 3}, {6, 2}}, {{5, 2}, {5, 1}}, {{6, 2}, {7, 2}}, {{7, 2}, {7, 1}}, {{5, 1}, {5, 0}}, {{7, 1}, {7, 0}}, {{5, 0}, {6, 0}}, {{6, 0}, {7, 0}}}]}, ImageSize -> 200], Graphics[{{Polygon[{{1, 7}, {1, 8}, {0, 8}, {0, 7}}], Polygon[{{8, 7}, {8, 8}, {7, 8}, {7, 7}}], Polygon[{{1, 5}, {1, 6}, {0, 6}, {0, 5}}], Polygon[{{2, 5}, {2, 6}, {1, 6}, {1, 5}}], Polygon[{{3, 5}, {3, 6}, {2, 6}, {2, 5}}], Polygon[{{4, 5}, {4, 6}, {3, 6}, {3, 5}}], Polygon[{{5, 5}, {5, 6}, {4, 6}, {4, 5}}], Polygon[{{1, 0}, {1, 1}, {0, 1}, {0, 0}}], Polygon[{{8, 0}, {8, 1}, {7, 1}, {7, 0}}]}, Line[{{{0, 0}, {0, 8}, {8, 8}, {8, 0}, {0, 0}}, {{1, 8}, {2, 8}}, {{1, 8}, {1, 7}}, {{2, 8}, {3, 8}}, {{3, 8}, {3, 7}}, {{0, 7}, {1, 7}}, {{0, 7}, {0, 6}}, {{3, 7}, {3, 6}}, {{0, 6}, {1, 6}}, {{1, 6}, {2, 6}}, {{2, 6}, {3, 6}}, {{3, 8}, {4, 8}}, {{3, 8}, {3, 7}}, {{4, 8}, {5, 8}}, {{5, 8}, {6, 8}}, {{6, 8}, 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3}, {4, 2}}, {{3, 2}, {3, 1}}, {{4, 2}, {4, 1}}, {{3, 1}, {3, 0}}, {{4, 1}, {5, 1}}, {{5, 1}, {6, 1}}, {{6, 1}, {6, 0}}, {{3, 0}, {4, 0}}, {{4, 0}, {5, 0}}, {{5, 0}, {6, 0}}, {{5, 3}, {6, 3}}, {{5, 3}, {5, 2}}, {{6, 3}, {6, 2}}, {{4, 2}, {5, 2}}, {{4, 2}, {4, 1}}, {{6, 2}, {7, 2}}, {{7, 2}, {7, 1}}, {{4, 1}, {5, 1}}, {{5, 1}, {6, 1}}, {{6, 1}, {6, 0}}, {{7, 1}, {7, 0}}, {{6, 0}, {7, 0}}}]}, ImageSize -> 200], Graphics[{{Polygon[{{1, 7}, {1, 8}, {0, 8}, {0, 7}}], Polygon[{{8, 7}, {8, 8}, {7, 8}, {7, 7}}], Polygon[{{1, 5}, {1, 6}, {0, 6}, {0, 5}}], Polygon[{{2, 5}, {2, 6}, {1, 6}, {1, 5}}], Polygon[{{3, 5}, {3, 6}, {2, 6}, {2, 5}}], Polygon[{{4, 5}, {4, 6}, {3, 6}, {3, 5}}], Polygon[{{5, 5}, {5, 6}, {4, 6}, {4, 5}}], Polygon[{{1, 0}, {1, 1}, {0, 1}, {0, 0}}], Polygon[{{8, 0}, {8, 1}, {7, 1}, {7, 0}}]}, Line[{{{0, 0}, {0, 8}, {8, 8}, {8, 0}, {0, 0}}, {{1, 8}, {2, 8}}, {{1, 8}, {1, 7}}, {{2, 8}, {2, 7}}, {{0, 7}, {1, 7}}, {{0, 7}, {0, 6}}, {{2, 7}, {3, 7}}, {{3, 7}, {4, 7}}, {{4, 7}, {4, 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4}, {2, 4}}, {{1, 4}, {1, 3}}, {{3, 4}, {4, 4}}, {{4, 4}, {4, 3}}, {{1, 3}, {2, 3}}, {{2, 3}, {2, 2}}, {{3, 3}, {4, 3}}, {{3, 3}, {3, 2}}, {{2, 2}, {3, 2}}, {{3, 5}, {4, 5}}, {{3, 5}, {3, 4}}, {{4, 5}, {5, 5}}, {{5, 5}, {6, 5}}, {{6, 5}, {6, 4}}, {{3, 4}, {4, 4}}, {{4, 4}, {4, 3}}, {{5, 4}, {6, 4}}, {{5, 4}, {5, 3}}, {{4, 3}, {4, 2}}, {{5, 3}, {5, 2}}, {{4, 2}, {5, 2}}, {{7, 5}, {8, 5}}, {{7, 5}, {7, 4}}, {{8, 5}, {8, 4}}, {{7, 4}, {7, 3}}, {{8, 4}, {8, 3}}, {{6, 3}, {7, 3}}, {{6, 3}, {6, 2}}, {{8, 3}, {8, 2}}, {{6, 2}, {7, 2}}, {{7, 2}, {7, 1}}, {{8, 2}, {8, 1}}, {{7, 1}, {8, 1}}, {{1, 3}, {2, 3}}, {{1, 3}, {1, 2}}, {{2, 3}, {2, 2}}, {{1, 2}, {1, 1}}, {{2, 2}, {2, 1}}, {{1, 1}, {1, 0}}, {{2, 1}, {3, 1}}, {{3, 1}, {4, 1}}, {{4, 1}, {4, 0}}, {{1, 0}, {2, 0}}, {{2, 0}, {3, 0}}, {{3, 0}, {4, 0}}, {{3, 3}, {4, 3}}, {{3, 3}, {3, 2}}, {{4, 3}, {4, 2}}, {{2, 2}, {3, 2}}, {{2, 2}, {2, 1}}, {{4, 2}, {5, 2}}, {{5, 2}, {5, 1}}, {{2, 1}, {3, 1}}, {{3, 1}, {4, 1}}, {{4, 1}, {4, 0}}, {{5, 1}, {5, 0}}, {{4, 0}, {5, 0}}, {{5, 3}, {6, 3}}, {{5, 3}, {5, 2}}, {{6, 3}, {6, 2}}, {{5, 2}, {5, 1}}, {{6, 2}, {7, 2}}, {{7, 2}, {7, 1}}, {{5, 1}, {5, 0}}, {{7, 1}, {7, 0}}, {{5, 0}, {6, 0}}, {{6, 0}, {7, 0}}}]}, ImageSize -> 200]} I cannot imagine that this pentomino puzzle, that has been solved independently by so many people, including myself and a former colleague of mine, and who all found 2170 solutions, in fact would have more than 2170 solutions. Moreover, the 8 extra solutions are not found by the uncompiled function. (Is that a bug?) But when the solutions are not new, it must be possible to transform 8 of the above solutions by rotating and reflecting into 7 others. Maybe that is obvious, but I fail to see how that can be done. Any comment will be highly appreciated. Fred Simons Eindhoven University of Technology