Will a polyomino of size less than 25 that does not surround a 1x1 hole be discovered by December 1, 2022?
Resolved
NO

In this Math StackExchange post, I asked about the existence of polyominoes that cannot be placed in a non-overlapping fashion to cover all four sides of an uncovered central unit square. The smallest solution currently known has 25 cells, and is shown below:

On the other hand, the best known lower bound is only 15 cells. It is an open question what the optimal solution is, and I'm fairly agnostic as to the possibility for improvement on the above shape.

This market resolves to YES if, by December 1 2022, I have confirmed the existence of a smaller polyomino than the one above that does not weakly surround a 1x1 hole.

(The most likely route through which I expect such a polyomino to be found is by someone discovering a new solution for the sake of trading on that information in this market, and then publicizing the result in the comments; I don't otherwise have reason to expect progress on this problem in the near future, and don't plan to work on it myself before the resolution date.)

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juan avatar

I'm trying to understand the specification better; when "placing a polyomino", are we allowed to perform rigid transformations (i.e. rotation, reflection) on copies of the polyomino?

maskedtorah avatar
Drake Thomas
is predicting NO at 12%

@juan Yes, although from some brief experimentation it seems like the question is still nontrivial with just translations so counterexamples for the translation case might still be promising to explore. (Every polyomino with up to 12 cells can surround a 1x1 hole via translation.)

AngolaMaldives avatar
Angola Maldives
bought Ṁ10 of NO

Voting NO as an ante for discovery.

maskedtorah avatar
Drake Thomas
bought Ṁ20 of NO

I've wagered some money on NO, partly because I expect by default this won't get enough attention for a really good search and partly because I'd like would-be discoverers to have a higher payout for doing so.

ManifoldDream avatar

Will a polyomino of size less than 25 that does not surround a 1x1 hole be discovered by December 1, 2022?, 8k, beautiful, illustration, trending on art station, picture of the day, epic composition