#sat-solving

Can the integers $1,2,\dots ,N$ be colored red and blue so that no Pythagorean triple $a^2+b^2=c^2$ has $a$, $b$ and $c$ all the same color? Find the largest $N$ for which this is possible, or show that it is possible for every $N$.

Keller conjectured in 1930 that in every tiling of ${\mathbb{R}}^n$ by unit cubes, some two cubes share an entire $(n-1)$-dimensional face. For which dimensions $n$ is this true?

A set of points in the plane is in general position if no three lie on a line. An empty hexagon in such a set is a convex hexagon whose vertices are points of the set and which contains no other point of the set. Find the smallest $N$ such that every $N$ points in general position contain an empty…