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As an example of the type of problem, consider Stewart Coffin's Cruiser puzzle:

Let R be a 48 × 31 rectangle. Let T be a 30°-60°-90° triangle with hypotenuse 34.565 (so legs are 17.2825 and 29.934). Let P be a right trapezoid with base 20.0982 and altitudes 27.1178 and 15.514. Show how to place two copies of T and two copies of P inside R with no overlap.

Pieces for the cruiser puzzle Solution

The algorithm should be able to solve for any number of polygons of any shape within another polygon of any shape (or conclude that such a solution is not possible).

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Probably not.

The bin-packing problem is NP-complete, and any instance of the bin-packing problem can be turned into a 2D packing problem. (Represent each integer $k$ with a $1\times k$ rectangle, and make the bins of size $n$ into several $1\times n$ holes with tiny "channels" connecting them into one full shape.) Therefore the 2D packing problem is at least as hard as the bin-packing problem.

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