Ever See A Tree in a Movie? Trees for movies are being outsourced to SpeedTree now. Print 3D Puzzles Some discussion of Oskar van Deventer and George Miller is in the article The Puzzle Masters. There is no way to match the three holes to the cluster's even number of perimeter holes, QED." But three of these bases have holes, and three do not. Call the edge opposite this hole the "base" these bases ought to match up to the cluster's six edges. Now consider the six triangles with one corner hole. These four tiles have six holes among their edges, and an even number of these must be internal, leaving an even number (two or four) for the perimeter. With that fiddly part out of the way, note that in either case, the four solid-corner tiles form a single cluster with six perimeter edges and six internal edges in three pairs. This can't be the C shape, as that would make five faces with no corner holes, but the other two shapes are fine. Repeating this logic shows that three, then all four triangles with solid corners are connected in one tetriamond. But since each vertex has only five neighboring vertices, then whether or not the two triangles just mentioned share an edge, there must be two triangles that actually do share an edge. By the Pigeonhole Principle, we can say that at least one vertex belongs to two or more triangles. Note the four tiles have twelve corners total. There are four tiles with no corner holes these must share the six no-hole vertices, and can do so in one of two ways. With 2-sided rotational symmetry, there are 20 ways - can they make an icosahedron with edge matching? Bryce Herdt: "First, each vertex must be surrounded by either 0 or 5 corner holes, and there must be six of each vertex. They fit nicely into a hexagon with matching.
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