> Thank you all for an easy-to-understand puzzle. > Mark's point that a plane tiling has no orientation > seems to me to make his identifying rotations of the > wallpaper a reasonable simplification of the original > problem, as does Avni's separating even n from odd. > > What about starting directly with the number of > possible patterns? Maybe that's what you're doing, > but not what I understood. This could be done with > the original problem or with Mark's modification, or > further simplification (no longer tiling)such as > reflexions, and even "inversions" (maybe not the > right word) where black and white interchange gives > what can be called the same pattern. > > Or going the other way, we could add a restriction > which brings the pattern even closer to real > wallpaper and further from tiling (as on a floor), by > confining the pattern within vertical edges. Thus > alternating vertical lines of black and white would > be different from vertical lines of white and black > or from odd numbers of stripes, as on a roll of > old-fashioned wallpaper! What if we started with an m > by n rectangle where m and n are not necessarily > equal?
Welcome to Rouben's challenge. I'm sure you can help. Rouben's comment (elsewhere in this thread) about primes being different instead of odd vs. even is interesting. It explains why both Avni's formula and my formula for odd numbers fit at N=2. I think Rouben states in his first post that the formula for total possible checkerboards is 2^(N^2).