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MathGroup Archive 2010

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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, 3}},
            {{4, 4}, {5, 4}}, {{5, 4}, {5, 3}}, {{3, 3}, {3, 2}},
            {{4, 3}, {5, 3}}, {{4, 3}, {4, 2}}, {{2, 2}, {3, 2}},
            {{2, 2}, {2, 1}}, {{4, 2}, {4, 1}}, {{2, 1}, {3, 1}},
            {{3, 1}, {4, 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}}, {{4, 3}, {5, 3}}, {{4, 3}, {4, 2}},
            {{5, 3}, {6, 3}}, {{6, 3}, {6, 2}}, {{4, 2}, {4, 1}},
            {{5, 2}, {6, 2}}, {{5, 2}, {5, 1}}, {{4, 1}, {4, 0}},
            {{5, 1}, {6, 1}}, {{6, 1}, {6, 0}}, {{4, 0}, {5, 0}},
            {{5, 0}, {6, 0}}, {{6, 3}, {7, 3}}, {{6, 3}, {6, 2}},
            {{7, 3}, {7, 2}}, {{5, 2}, {6, 2}}, {{5, 2}, {5, 1}},
            {{7, 2}, {8, 2}}, {{8, 2}, {8, 1}}, {{5, 1}, {6, 1}},
            {{6, 1}, {6, 0}}, {{7, 1}, {8, 1}}, {{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}, {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}},
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            {{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}}, {{2, 4}, {3, 4}},
            {{3, 4}, {3, 3}}, {{4, 4}, {5, 4}}, {{5, 4}, {5, 3}},
            {{3, 3}, {3, 2}}, {{4, 3}, {5, 3}}, {{4, 3}, {4, 2}},
            {{3, 2}, {4, 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}}, {{1, 4}, {2, 4}}, {{1, 4}, {1, 3}},
            {{2, 4}, {3, 4}}, {{3, 4}, {3, 3}}, {{1, 3}, {2, 3}},
            {{2, 3}, {2, 2}}, {{3, 3}, {3, 2}}, {{2, 2}, {2, 1}},
            {{3, 2}, {4, 2}}, {{4, 2}, {4, 1}}, {{2, 1}, {3, 1}},
            {{3, 1}, {4, 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}}, {{4, 3}, {5, 3}}, {{4, 3}, {4, 2}},
            {{5, 3}, {6, 3}}, {{6, 3}, {6, 2}}, {{4, 2}, {4, 1}},
            {{5, 2}, {6, 2}}, {{5, 2}, {5, 1}}, {{4, 1}, {4, 0}},
            {{5, 1}, {6, 1}}, {{6, 1}, {6, 0}}, {{4, 0}, {5, 0}},
            {{5, 0}, {6, 0}}, {{6, 3}, {7, 3}}, {{6, 3}, {6, 2}},
            {{7, 3}, {7, 2}}, {{5, 2}, {6, 2}}, {{5, 2}, {5, 1}},
            {{7, 2}, {8, 2}}, {{8, 2}, {8, 1}}, {{5, 1}, {6, 1}},
            {{6, 1}, {6, 0}}, {{7, 1}, {8, 1}}, {{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}, {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}},
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            {{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}}, {{5, 5}, {6, 5}}, {{6, 5}, {6, 4}},
            {{7, 5}, {7, 4}}, {{5, 4}, {6, 4}}, {{5, 4}, {5, 3}},
            {{7, 4}, {7, 3}}, {{5, 3}, {6, 3}}, {{6, 3}, {7, 3}},
            {{0, 5}, {1, 5}}, {{0, 5}, {0, 4}}, {{1, 5}, {2, 5}},
            {{2, 5}, {2, 4}}, {{0, 4}, {0, 3}}, {{1, 4}, {2, 4}},
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            {{0, 2}, {0, 1}}, {{1, 2}, {1, 1}}, {{0, 1}, {1, 1}},
            {{2, 5}, {3, 5}}, {{2, 5}, {2, 4}}, {{3, 5}, {3, 4}},
            {{1, 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}},
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            {{4, 3}, {4, 2}}, {{5, 3}, {5, 2}}, {{4, 2}, {5, 2}},
            {{7, 5}, {8, 5}}, {{7, 5}, {7, 4}}, {{8, 5}, {8, 4}},
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            {{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}},
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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}}],
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            {{7, 4}, {7, 3}}, {{5, 3}, {6, 3}}, {{6, 3}, {7, 3}},
            {{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}, {3, 4}},
            {{1, 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}},
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            {{5, 2}, {5, 1}}, {{6, 2}, {7, 2}}, {{7, 2}, {7, 1}},
            {{5, 1}, {5, 0}}, {{7, 1}, {7, 0}}, {{5, 0}, {6, 0}},
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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}}],
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            {{8, 7}, {8, 6}}, {{4, 6}, {5, 6}}, {{5, 6}, {5, 5}},
            {{6, 6}, {7, 6}}, {{6, 6}, {6, 5}}, {{7, 6}, {8, 6}},
            {{5, 5}, {6, 5}}, {{6, 6}, {7, 6}}, {{6, 6}, {6, 5}},
            {{7, 6}, {7, 5}}, {{4, 5}, {5, 5}}, {{4, 5}, {4, 4}},
            {{5, 5}, {6, 5}}, {{7, 5}, {7, 4}}, {{4, 4}, {5, 4}},
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            {{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, 6}}, {{0, 6}, {1, 6}},
            {{1, 6}, {2, 6}}, {{2, 6}, {3, 6}}, {{3, 6}, {4, 6}},
            {{2, 8}, {3, 8}}, {{2, 8}, {2, 7}}, {{3, 8}, {4, 8}},
            {{4, 8}, {5, 8}}, {{5, 8}, {5, 7}}, {{2, 7}, {3, 7}},
            {{3, 7}, {4, 7}}, {{4, 7}, {4, 6}}, {{5, 7}, {6, 7}},
            {{6, 7}, {6, 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}}, {{5, 5}, {6, 5}}, {{6, 5}, {6, 4}},
            {{7, 5}, {7, 4}}, {{5, 4}, {6, 4}}, {{5, 4}, {5, 3}},
            {{7, 4}, {7, 3}}, {{5, 3}, {6, 3}}, {{6, 3}, {7, 3}},
            {{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}, {3, 4}},
            {{1, 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


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