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