Base b is "robust" if each fraction 1/2,1/3,..,1/(b-1) is a ratio of two integers expressed in base b that use each digit 1, 2, 3,.... (up to some value) exactly once. (1/b cannot be so expressed. Why?) @iconjack shows base 10 is robust. Base 3 and base 4 are robust. Base 5? 6?

N>1 cups all upright. A "move" consists of turning all but 2 cups over. (So, each move N-2 cups change state upright/upside-down.) For which N is it possible to make all N cups upside-down? [General theory about N cups and turning N-k over at a time?]

Square tiles in an NxM grid with one blank space. Tiles can slide H and V into the space. Starting with a blue tile top right, blank space to its left ... For which N and M is possible to make slides and have the blue tile visit each and every location in the grid?

An inf number of bags. First contains one Red & one White ball. Second 2R & 1W balls. Third 3R & 1W balls. I pull out a ball from first bag. If R, I win! If not, I go to second bag & pull out a ball. If R, I win! If not, on to next bag, and so on. Will I win? Expected # of turns?

This configuration of 16 triangles has perimeter 14. (Assume each triangle side length is 1.) Remove triangles one at a time, always increasing perimeter. What's the least number of triangles you can get to? What is the final total perimeter of the configuration you have?

Suppose you find n consecutive integers that sum to a square number r^2. Prove that those n integers plus the next r integers added together equals the sum of the following r integers. Moreover, prove that n integers after all that sum to (r+n)^2 so that this process repeats.