Can points on an integer lattice form the vertices of a regular hexagon Is it possible for six points in the integer lattice to form the vertices of a regular hexagon

No. By Pick's Theorem, the area of such a hexagon is rational (in fact, it is an integer multiple of $\frac 12$).
But by an alternative geometric method, you can show that the area of a regular hexagon is $A = \frac{3\sqrt 3}{2}a^2$, where $a$ is the side length. The square of the side length of a lattice polygon is integral by the Pythagorean theorem, so $A$ is irrational, and you have a contradiction.
