POLYOMINOES, UNSOLVED PROBLEMS
The problems below are adapted from my book POLYOMINOES,
An Introduction to Puzzles and Problems in Tiling, published
by the Mathematical Association of America.
The 1996 reprint
has an Addenda announcing that each of Problems #3.11, #5.22,
and #5.27 has been solved by showing that the desired
construction turns out to be impossible.
The remaining
problems that were listed as unsolved remain unsolved, as far
as I know. Some, such as #1.4, are expected to remain
unsolved for some time. Others may yield to a computer
attack. Those from Chapter 9 are of special interest to the
author. (An n-omino consists of n rookwise connected squares
from an infinite checkerboard. A polyomino is p-poic if
directly congruent copies tile the plane in exactly p
noncongruent ways. A polyomino is m-morphic if congruent
copies tile the plane in exactly m noncongruent ways.)
Please let me know if you solve any of these problems.
Email address: martin@math.albany.edu
Best wishes!
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#1.4. Give a formula for the number of n-ominoes for positive
integers n.
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#3.12. Does there exist a polygon and an odd integer k with
k>1 such that k is the smallest number of copies of the
polygon needed to tile a rectangle?
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#4.13. In how many ways can an n by n square be tiled with
square polyominoes?
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#4.14. In how many ways can an h by w rectangle be tiled with
an a by b rectangular polyomino?
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#4.15. In how many ways can an h by w rectangle be tiled with
rectangular polyominoes?
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#5.4. In forming an 8 by 8 square with the twelve pentominoes
and two dominoes, where can the dominoes be placed within the
8 by 8 square?
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#6.10. Is there a polyomino reptile that does not tile a
rectangle?
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#7.11. Other than the P-hexomino, is there a nonrectangular
polyomino such that n copies form a rectangle where n is odd
and n<12?
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#7.14. Is the hexomino below odd?
s
s s s s s
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#8.13. What is the smallest odd integer n such that a square
can be cut into n congruent nonrectangular polygons?
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#8.15. For which positive integers n does there exist some
nonrectangular polygon such that n copies of the polygon form
a rectangle?
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#8.16. For which positive integers n does there exist some
polygon such that n is the minimum number of copies required
to form a rectangle?
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#9.9. (There are five known families of trimorphic Z's.) Find
some more trimorphic Z's.
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#9.16. Find another decamorphic polygon.
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#9.17. Find an octapoic polygon.
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#9.18. Find a p-poic polygon for some p such that p>8.
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#9.19. Find a hendecamorphic polygon.
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#9.20. Find a m-morphic polygon for some m such that m>11.
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December 30, 1997