Posts Tagged ‘tiling’

Pentomino Layer Cake

February 27th, 2010

On the Polyforms list, Erich Friedman posed a very interesting new pentomino tiling problem:

Tile a rectangle of minimal area with pentominoes so that for each pentomino there is exactly one stratum, or cluster of one or more copies of that pentomino that reaches from one side of the rectangle to the opposite side. Pentominoes in a stratum must form a single group, connected by edges, not just corners.

Michael Reid found this 3×30 solution:

It’s not hard to prove that it is minimal. A natural extension of the problem is to find minimal solutions for 4×n and 5×n rectangles. Michael Reid found the first 5×n solution, but I improved on it with this 5×32 solution:

The 4×n problem seems to be the hardest, and initially it was not clear that it would be possible. The X pentomino has only one possible stratum, which only can only be bordered by Y, I or N, and it is also difficult to find matches for a Z stratum. Additionally, only Y, L, and P can form straight line stratum boundaries usable for the top and bottom of the rectangle. (See wikipedia’s pentomino page if you don’t know the correspondence between letters and shapes.) I did eventually find this 4×50 solution:

This solution seems rather prolifigate with its pentominoes, but finding any solution at all was a bit of a surprise.

Update: Erich Friedman’s Math Magic for April 2010 further explored this subject.

Hexiamonds on an Octahedron

January 26th, 2010

Here’s an interesting problem that seems not to have gotten as much attention as I think it deserves. The twelve hexiamonds contain a total of 72 triangular units. A regular octahedron with edges 3 units long can fit 9 triangles on each of its 8 faces, exactly enough to tile with the hexiamonds. Some individual solutions to this problem have been found. A solution at Livio Zucca’s site bears the label “Adrian Struyk 1963?” so we may assume the problem has been around at least since then. Another solution by Michael Dowle is here.

Notice that you can unfold an octahedron to produce a net in the form of an octiamond. This provides another source for solutions. The octahedron has 11 different octiamond nets. Pieter Torbijn found that 24 enlarged octiamonds could be tiled with the hexiamonds; of these, 5 are nets of an octahedron.

As far as I know, nobody has made an exhaustive computerized search for solutions to this problem. You can be the first!

#4: How many distinct ways can the hexiamonds tile an octahedron?

The octahedron has a large amount of symmetry compared to any planar figure that these pieces can tile. It has 48 automorphisms, or ways to map the solid onto itself. This would indicate a relatively small number of different solutions, since many solutions will be mappings of each other over the various ways of rotating and reflecting the octahedron. On the other hand, the shape lacks external borders, which ought to greatly increase the number of possibilities.

Some solutions have a piece that wraps around and caps a vertex. This could be considered an aesthetic flaw, because it would be impossible to tell which hexiamond the capping piece is just from knowing what triangles it occupies; you must also know its edges.

There are 7 different pieces that can cap a vertex, one of which, the pistol, can cap it in two distinct ways. Notice that due to the symmetry of the shape it makes when it caps a vertex, only the orientation of the sphinx is a riddle; its identity is never in question.

#5: How many solutions have a capped vertex?

#6: What is the largest number of vertices that can be capped in a solution? The ideal would be for all six vertices to be capped with all of the above hexiamonds except the sphinx.

The octahedron has twelve edges, the same as the number of hexiamonds. This suggests another problem:

#7: Is it possible to tile the octahedron so that each of the twelve hexiamonds is folded across exactly one of the edges of the octahedron?