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Math Magic Archive

Problem of the Month (December 2018)

This month we investigate miscellaneous polyomino problems. We say P ⊆ Q if P can be packed inside Q. We say P ⊂ Q if P ⊆ Q and P ≠ Q.

1.

Fix a polyomino P. We say a finite polyomino Q is critical if P ⊄ Q, and Q ⊂ R implies P ⊂ R. In other words, P won't fit inside Q, but if you add any square to Q then P will fit. Given P, is there a critical Q? If so, what is the smallest such Q? Is there a largest such Q, or are there arbitrarily large Q? What if P is a collection of polyominoes, none of which fit in Q, but any larger polyomino contains some polyomino from P?

2.

Fix a n-omino P. Is there an infinite polyomino Q with the property that the only n-omino P with P ⊂ Q is P? What if P is a collection of polyominoes, which are the only n-ominoes that are subsets of Q?

3.

What is the smallest polyomino that can be surround by exactly n copies of itself, with each of those copies being surrounded by exactly m copies of itself? Here "surrounded" means all the edges, but not necessarily the corner points.

4.

How many ways are there to tile a rectangle with a set of polyominoes? In general, this is a hard problem. But the problem of how many ways an n×k rectangle can be tiled for fixed k with a fixed set of polyominoes can be solved by recurrence relations. For example, the number of ways an an n×1 rectangle can be tiled with 1×1 and 2×1 rectangles clearly satisfies the recurrence formula an = an-1 + an-2. What are the recurrence formulas for other small cases?

5.

Given a polyomino, what is the shortest linear absolute value inequality in two variables whose solution set is that polyomino? Are such inequalities always possible?


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