Rhombs with Differently Colored Sides

When I filled polygons with rhombs I never tried to use rhombs with differently colored sides, because with these pieces we can't never get a polygon with uniformly colored border. At least at corners with sharp angles a piece with differently colored sides contributes different colors to the two legs of the angle. When you cover polyrhons instead there is no border at all and no problem with differently colored pieces.

What about a single rhon? Since a single rhon has 12 faces the 12 different rhombs using 4 colors might cover it. But this is impossibe. 12*4/4/2=6 edges of same color must be distributed so that they aren't sides of one rhomb and this can't be done. Therefore I looked for constructions with n>4 colors.

5 Colors, 60 Pieces

Here are the 60 pieces shown as a rhonbic ring with four uniformly colored sides and an uniformly colored center hole. A SVG-file is also provided.

The hexagonal ring and the hexagon made from six rhons can be covered by the set. A physical construction of the ring is shown in the title.

Most hexarhons have 6*10+2=62 faces, but there are also some other hexarhons with two additional faces, which are connected, giving a surface of 60 faces. Two pieces in the orthogonal grid, looking like hexominoes, are shown.

Here are two non planar polyrhons with 60 faces.

Five single rhons with no touching faces can also be covered.

6 Colors, 180 Pieces

When I noticed that a tertahedron like construction of size 5 has exactly 180 rhombic faces, I tried to cover them with the set. The order to cover the faces must be carefully determined.

Chains of 18 rhons have 18*10+2=182 faces, but if the chain is closed like a ring we get only 180 faces.

7 Colors, 420 Pieces

An arrangement of n x n rhons in the orthogonal grid has n*8+4*n faces. Since 7*7*8+4*7=420 I tried to find a solution. During the first stage of the algorithm only pieces with a fixed color at one side had to be used.

8 Colors, 840 Pieces

A square of size 10 x 10 can also be covered.

With better algorithms, faster computers and more time larger constructions might be possible, too.

1 2 3 k
Colors 4 7 8 11 12 15 4k+1 4k+3
Pieces 12 420 840 3960 5940 16380 (4k+1)!/(4k-3)!/2 (4k+3)!/(4k-1)!/2
Size of Square 1 7 10 22 27 45 4k^2-3k 4k^2+3k

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