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Geometric Puzzles in the Classroom

Simultaneous Pentomino Rectangles

by Henri Picciotto
Simultaneous 3x5 and 5x9
This article is reprinted, with permission, from Michael Keller's games and puzzles 'zine (#9, Dec 1989).

For a little background and context on pentominoes, see
Geometric Puzzles in the Classroom.

I came up (by hand) with examples of most pairs of simultaneous pentomino rectangles. They can be found in my book Pentomino Activities, Lessons, and Puzzles (Where to get it.). Some combinations (or the fact that they were impossible) could be found in the original computer searches concerning the 6x10, 5x12, and 4x15 rectangles. Others remained unsolved, at least to my knowledge. The full solution had to wait for a complete computer search, which has now been completed by Ted Hertz (see accompanying article).

It is interesting to study the table of pentomino rectangle pairs:

  1. Of the 46 pairs, all but 13 are possible.
  2. The 2x10 accounts for nine of the impossible pairs. It can only be paired with the 3x10.
  3. The 1x5 (the pentomino I) can be paired with every rectangle except the 2x10.
  4. Three puzzles have unique solutions: the devilishly difficult 5x5/5x7 and 5x3/5x9, and the easy 2x10/3x10.
  5. The rectangle with the most individual solutions is the 5x10.
  6. Excluding the pairs with 1x5, the pair with the most solutions by far is 4x5/6x5.

Here is an interesting challenge: can you pair the LNPU 4x5 with any other rectangle except 1x5? I do not know whether it is possible.

Three simultaneous pentomino rectangles cannot be achieved if one excludes the 1x5 rectangle. If there were three rectangles, they would have to be one of the following eleven cases:

3x5   3x5   3x5   3x5   3x5   3x5   3x5   3x5   3x5   4x5   4x5 
3x5   3x5   3x5   3x5   3x5   3x5   4x5   4x5   4x5   4x5   4x5 
3x5   4x5   2x10  5x5   6x5   3x10  4x5   2x10  5x5   4x5   2x10
 a     a'    a     b'    b     b     a'    a     b"    b     b  

I found by hand all the solutions to pairs of the three basic types: 3x5/3x5, 3x5/4x5, and 4x5/4x5. Such pairs must be a subset of all potential triples. These pairs were analyzed and the eleven cases divided into five categories:

a -- the two rectangles between them use the L, P, and U pentominoes, which makes the third rectangle impossible.

a' -- the same as a, but there is a unique third rectangle (NTVY) which does not use L, P, or U, and must be ruled out individually.

b -- the two rectangles do not use the X, which must be used in the third (but can't be).

b' -- the two rectangles do not use X or I, which must both be used in the third (but can't be).

b" -- the two rectangles do not use the X, which must be used in the third. But the third would then need the P also, which is used in one of the first two.

Though I am confident that this analysis is correct, I realize that this sort of approach lends itself to error. Fortunately, the result was confirmed by Ted Hertz's computer search.

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Multiple Pentomino Rectangles

by Ted Hertz
Simultaneous 5x5 and 5x7
This article is reprinted, with permission, from Michael Keller's games and puzzles 'zine (#9, Dec 1989).

Solutions were obtained using a Commodore 128 computer. An article in Compute! magazine entitled "Pentominoes -- A Puzzle Solving Program" by Jim Butterfield (May 1984, pp. 106-122) was modified and translated from Basic to 6502 machine language, improving the solution speed by a factor of approximately 1000 to 1.

The number of solutions to each rectangle is given in the first row of the table, labeled 'Alone'. Pairs of which the 1x5 rectangle it is a member are easily found by performing a search of rectangle solutions for all those not containing the I pentomino. These numbers appear in the second row, labeled '1x5'.

For the remaining rectangle pair solutions, exhaustive searches were performed between members of the rectangle solution list for pairs having no pentomino in common. For those cases in which both rectangles had the same dimensions, programming care was exercised to avoid duplicate solutions. By comparing the lists of rectangle pairs obtained above with all single rectangle solutions, it was determined that no three-rectangle solutions are possible except ones involving the 1x5 rectangle.

Pentomino Rectangle Pairs
       3x5 4x5 2x10  5x5  6x5 3x10  7x5  8x5  4x10   9x5 3x15  10x5  11x5
Alone   7   50   2   107  541  145 1396 3408  2085  5902  201  6951  4103
 1x5    7   36   0    35  205   82  398  775   621   780   19   416   112
 3x5    5   29   0    25   67   10   84   29     9     1    0            
 4x5        28   0    60  133   25   22    0     5                       
2x10             0     0    0    1    0    0     0                       
 5x5                  12   20    6    1                                  
 6x5                        2    0                                       
3x10                             0                                       
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Geometric Puzzles in the Classroom
Visit Henri Picciotto's Math Education Page. Send me e-mail .