# The hardest riddle on earth: Covering an 8×8 grid with trominoes

I already sought answers on three different forums without success. I hope I will be lucky this time.

I tried to find non-trivial solutions on a deficient 8×8 grid covered with trominoes and I regret to say that after extensive efforts I found only two non-trivial solutions:

diagram: 2 non-trivial solutions

The conditions of covering such a grid with trominoes are the following: In total we have 21 L-shaped trominoes with 3 different colors. There are equal numbers of trominoes of each color. Placing the trominoes on the grid, no 2 trominoes of the same color are allowed to touch each other anywhere, except only once, corner to corner (highlighted by red lines in the diagram).

Solutions from rotations and reflections are trivial. Does anyone know how to obtain more non-trivial solutions? Please include the full 8X8 grid in your answers.

It is more appropriate to be solved with a computer program. If you prefer to do it by hands, here is a workable procedure:

1. Place the hole. After removing symmetry redundancy, there are 10 distinct positions of the hole, namely: A1, B1, B2, C1, C2, C3, D1, D2, D3, D4;

2. Layout bricks, disregard color, you may want to work around the holes and spread out. Any partial layout that has additional holes in it is not feasible and should be discarded;

3. Color all feasible layouts obtained in step 2.

I was able to obtain a total of 49 unique solutions. Here is some examples:

01122113
33123133
13233221
11221211
23311332
22322322
31123113
33133133
solution:1 with corner touch:1

01122112
33123122
13233233
11221223
23311311
22322331
31123122
33133112
solution:2 with corner touch:1

01122122
33123112
13233233
11221223
23311311
22322331
31123122
33133112
solution:3 with corner touch:1

• You seem to have ignored this rule: "no 2 trominoes of the same color are allowed to touch each other anywhere, except only once, corner to corner" – r.e.s. Apr 21 '18 at 0:42
• Sorry. Can you name the two trominoes that touch each other. Reference a block by any of its cell. Eg. C2 and D1. Thanks! – Lance Apr 21 '18 at 0:48
• $(B2,C3), (E2,F3), (F2,E3), ...$ – r.e.s. Apr 21 '18 at 0:54
• @Lance: You have corners touching at H2,G3 and C6, B7 and E6,F7. – john Apr 21 '18 at 0:57
• The rule would be less ambiguous if stated something like this: "no 2 trominoes of the same color are allowed to touch each other anywhere, except at most one pair of the same color is allowed to touch corner to corner" – r.e.s. Apr 21 '18 at 1:15