Nonce Equitaur 2 wrote: With 7 colors, there are 35 triplets. Twenty of them don't have a 7 -- MPQ only has powers on 6 colors. 123, 124, 125, 126, 127, 134, 135, 136, 137, 145, 146, 147, 156, 157, 167, 234, 235, 236, 237, 245, 246, 247, 256, 257, 267, 345, 346, 347, 356, 357, 367, 456, 457, 467, 567 One set of 7 of those triplets is 124, 137, 156, 235, 267, 346, 457. Any two of them shares exactly one number. That's a Fano plane.