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I have been puzzling over this for a few years and still have no answer.

How many ways can a solid object like a cube or a ball be cut into pieces such that it is possible to take apart and put back together (in the real world or in math talk where each movement is a rigid body motion - translation and/or rotation)?

Clearly, some notion of isomorphism equivalence classes (math talk again) would be necessary with categories depending on the number of separate pieces.

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I doubt that there is any practical categorization that would work for all possible puzzles. People have designed mechanical analogues for Turing engines. So it is probably feasible to design a finite 3D block puzzle with some k*n components (where k components are sufficient to construct a single Turing engine state), that would take a number of separate manipulations comparable to BB(n) to disassemble (where BB(n) is the busy-beaver number for the n-state Turing engine).

The busy-beaver number grows extraordinarily quickly as a function of n, so it should be possible to design such a puzzle for which there wouldn't be any practical way to determine if it was soluble.

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