Here a,b are integers. You can use squares of different sizes.

For example letting f(a,b) be this minimal number, then f(2,3)=3 and f(5,6)=5.

2. Given a closed polygonal path p in R^3 composed of unit segments, is there an immersed polygonal surface

whose faces are equilateral triangles of edge length 1, spanning p? [Solved! See Glazyrin and Pak's article ]

3. For which triples of integers {-p,0,q} can you represent every integer with a finite base-3 expansion using digits -p,0,and q?

For example {-1,0,1} and {-1,0,7} but not {0,1,2} or {-1,0,4}.

4. Tile a fixed triangle with triangles of equal area, meeting edge-to-edge. Is there an uncountable number of such tilings? (In other words, is every such tiling rigid?)

5. How many closed paths of length n in the square grid have the property that they have winding number zero around every face? Equivalently, for the standard generators a,b of the free group F=F[a,b] on two generators, what is the growth of the subgroup [[F,F],[F,F]]?

6. Let t be an irrational multiple of pi. Consider the map of the complex plane to itself: f(z) = exp(it)(z+2sin(t)) if Im(z)>0 and f(z) = exp(it)(z-2sin(t)) if Im(z)<0. Show that the map is integrable: almost every point is on a circle which, under an iterate of f, undergoes a rigid rotation.

7. Let S be the intersection of the middle-third and middle-half Cantor sets (both constructed in the unit interval). Find or prove the existence of an irrational number in S.