Open problems

1. What is the minimal number of squares needed to tile an aXb rectangle?
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.