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.
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
Using 4 colors without repetition (6 Squares)
The six squares are just enough to cover a single cube with matching colors at the edges. Apart from rotation and reflection there is only one solution, which can be uniquely finished after fixing two neighboring squares. I printed a cube with holes where the panels click into place.
Using 5 colors without repetition (30 Squares)
There are only two boxes with a surface area of 30 unit squares (3x3x1 and 1x1x7). They can be covered by the 30 pieces with matching colors at the sides of the squares.
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.
Using 6 colors without repetition (90 Squares)
We can cover three boxes with these pieces (1x1x22, 2x5x5, 3x3x6). Mixing the pieces and simple backtrackingng was rather slow for the 3x3x6 box, but after prefering pieces with high colors during the first part of the search procedure I got the solution rather fast. For physical constructions I bought snap cubes and printed the quadratic panels to fit the holes in the cubes. Here are the panels and two constructions.
You can also view these and some other constructions as virtual objects.
Using 7 colors without repetion (210 Squares)
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.
Using 3 colors with same colored edges allowed (24 Squares)
This set is known to fill a 4x6 rectangle with matching colors inside and same colored edges at the border. It's also possible to panel a 2x2x2 cube with these pieces. The 2x2x2 cube is a special octacube and we can also panel a heptacube, 28 hexacubes and all pentacubes.
For the 29 pentacubes we need only 20 or 22 pieces and it should be possible to solve these problems by hand.
Using 4 colors with same colored edges allowed (70 Squares)
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.
Using 5 colors with same colored edges allowed (165 Squares)
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.
Using 6 colors with same colored edges allowed (336 Squares)
Backtracking was supported by taking pieces with high color edges first.
Using 7 colors with same colored edges allowed (616 Squares)
You can find solutions for boxes with two small edges rather easily. For the other ones I chose the following method:
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