Recursive Interlocking Puzzles

Interlocking puzzles are very challenging geometric problems with the fascinating property that once we solve one by putting together the puzzle pieces, the puzzle pieces interlock with one another, preventing the assembly from falling apart. Though interlocking puzzles have been known for hundreds of years, very little is known about the governing mechanics. Thus, designing new interlocking geometries is basically accomplished with extensive manual effort or expensive exhaustive search with computers. In this paper, we revisit the notion of interlocking in greater depth, and devise a formal method of the interlocking mechanics. From this, we can develop a constructive approach for devising new interlocking geometries that directly guarantees the validity of the interlocking instead of exhaustively testing it. In particular, we focus on an interesting subclass of interlocking puzzles that are recursive in the sense that the assembly of puzzle pieces can remain an interlocking puzzle also after sequential removal of pieces; there is only one specific sequence of assembling, or disassembling, such a puzzle. Our proposed method can allow efficient generation of recursive interlocking geometries of various complexities, and by further realizing it with LEGO bricks, we can enable the hand-built creation of custom puzzle games.