Given nine Lattice Points in three dimensions, prove that there are at least two for which their midpoint is also a lattice point.
HintThink about the odd-even parity of the nine pointsSolution
There are eight different odd/even parity "signature" of a point. For example \((1,4,-2)\) has the signature (odd, even, even). Since we are given nine points, the pigeon-hole principle guarantees us that there must be two different points with the same signature. The midpoint of two lattice points with the same signatures is a lattice point since \(\frac{odd + odd}{2}\) and \(\frac{even + even}{2}\) are both integers.
