2020/03/16

Edge colored squares can be used to fill rectangles with matching colors inside and same colors at the border. Long ago these problems were stated and examined. Some solutions are shown at my section about edge-colored polygons. Now let's try to cover the surfaces of polycubes especially boxes. I looked at some sets of edge colored squares with repetition of colors forbidden or allowed . Click the numbers to get possible constructions.

Colors | 3 | 4 | 5 | 6 | 7 | n |

All edges different | - | 6 | 30 | 90 | 210 | n!/4 |

Repetition allowed | 24 | 70 | 165 | 336 | 616 | (n^4 + n^2 + 2n)/4 |

On a single image it's impossible to see the backside of a solution, therefore I decided to create some 3d objects, which can be turned around and changed in size.

To save the solutions I used the following format for text files. Each square, which contributes to the surface is defined by the x y z coordinates of its cube and the character U, D, R, L, F, B (Up, Down, Right, Left, Front, Back) for the face of the cube. The colors are listed counterclockwise starting at the bottom for F and R, at the top for B and L, at the back for D and at the front for U. A zip-file with the solutions is here.

Since the 1x1x7 box is a heptacube I looked at other heptacubes to cover them with these pieces, too. For 882 out of 1023 heptacubes all pieces must be used; for the other ones some pieces are left. With snap cubes all heptacubes can be made, and then the faces can be paneled with the colored squares.

Since the number of pieces is rather small all solutions were found rather fast. After this an obj-file for each paneled heptacube was calculated. So you can see the solution by choosing a number between 1 and 1023.

Enter number of heptacube 1..1023:

Some octacubes can also be covered by this set.

You can also view these and some other constructions as virtual objects.

There are four boxes, which can be paneled with these pieces 1x1x52, 3x3x16, 5x5x8 and 4x7x7. As with the six color set pieces with at least one high color edge should be used during the first part of the search procedure.

For the 29 pentacubes we need only 20 or 22 pieces and it should be possible to solve these problems by hand.

Because I haven't produced a physical set I only solved the 4 possible boxes and an open box, which can be covered by all pieces.

The 165 Pieces can't cover a polycube, because the surface area of a polycube is always even. Therefore I discarded the 5 pieces with all edges same colored and looked for boxes with a surface area of 160.

Backtracking was supported by taking pieces with high color edges first.

- The first 430 pieces must have at least one high color edge (6 or 7)
- Stop, when a partial of 612 was reached
- Keep the first 500 pieces of the partial fixed and try to finish the whole construction with mixing and backtracking

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