Hearn has proved that this puzzle belongs to a class of computational problems described as NP-complete,
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. He describes his results in a recent issue of the Mathematical Intelligencer.
In its starting configuration,
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, a TipOver puzzle has several vertical crates of various heights (1 x 1 x h) arranged on a square grid. A tipper—representing a person navigating the layout—stands on top of a particular starting crate. There is a special 1 x 1 x 1 red crate—the target—elsewhere on the grid,
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The tipper can topple any vertical crate that it is standing on, in any of the four compass directions, provided there's enough space for the crate to fall unobstructed and lie flat. The tipper can walk (or climb) along the tops of any crates that are adjacent,
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, even when they have different heights.
In the sample puzzle and solution shown below,
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, the numbers indicate the vertical height of each untoppled crate. The tipper starts on the purple crate. In the first move (top, second from left),
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, the tipper has moved to one of the green crates (3 units tall) and toppled it southward so that it ends up adjacent to a yellow crate that is 2 units tall.
"Surprisingly,
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, it does not take many crates to make an interesting puzzle," Hearn writes. "The number of tips required can never be more than the number of crates—once a crate has been tipped over, it stays fallen—yet finding the correct sequence can be quite a challenge."
To prove that TipOver is NP-complete,
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, Hearn showed that TipOver puzzles are related to a well-known problem

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