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# Covering a rectangle with pentominoes

## Question

There are 12 ways to form a pentomino from five squares.

Can you arrange them to cover a 3x20 rectangle?

Why or why not?

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In the OP, you asked why or why not.

Can the existence of a solution be verified before finding one?

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I think it could only be disproven.

As in removing opposite corners from a chessboard and tiling the result with dominos.

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A loosely related question: Is it possible to prove that a statement cannot be proven but can be disproven?

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Is this what you mean?

Any evidently false statement (A standard deck of cards has five aces.)

1. Is not true
2. Therefore cannot be proved to be true.
3. Which proves that it cannot be proved to be true
4. Can be disproved by counting the aces.

You disprove a statement by negating a necessary condition. You prove a statement by establishing a sufficient condition. So you can or you can't prove or disprove (something), depending on the existence and truth or falsity of necessary or sufficient conditions.

In the OP I don't know of sufficient conditions that can be shown to exist. k-man discovered a successful tiling, but I don't see how to prove a priori that one exists. In the case of tiling with dominoes a chessboard with opposite corners removed, there is a necessary condition (equal numbers of white and black squares) that does not exist. You know a priori that you can't do it.

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Is this what you mean?

Any evidently false statement (A standard deck of cards has five aces.)

• Is not true
• Therefore cannot be proved to be true.
• Which proves that it cannot be proved to be true
• Can be disproved by counting the aces.
You disprove a statement by negating a necessary condition. You prove a statement by establishing a sufficient condition. So you can or you can't prove or disprove (something), depending on the existence and truth or falsity of necessary or sufficient conditions.

In the OP I don't know of sufficient conditions that can be shown to exist. k-man discovered a successful tiling, but I don't see how to prove a priori that one exists. In the case of tiling with dominoes a chessboard with opposite corners removed, there is a necessary condition (equal numbers of white and black squares) that does not exist. You know a priori that you can't do it.

That's not what I meant, but what you said afterwards made a lot of sense. It brought order to the muddled up thoughts in my mind. I guess what I meant was, is it possible to prove that no sufficient conditions exist?

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is it possible to prove that no sufficient conditions exist?

You would have to enumerate them, individually or by classification, to do that.

Individually, by inspection or argument, conditions can be shown not to exist.

By classification, you could show that conditions of a particular type (appropriate to the statement) do not exist.

It could be done, but it could also be difficult.

You'd have to be certain that you analyzed them all.

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