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E6. Lattice triangles and tetrahedrons

In two dimensions, a lattice polygon is a polygon in a Cartesian coordinate plane such that the two coordinates of each vertex are integers. In three dimensions, a lattice polyhedron is a polyhedron such that the three coordinates of each vertex are integers.

(a) Prove that a lattice triangle cannot be an equilateral triangle.

(b) Is it possible for a lattice tetrahedron to be a regular tetrahedron?

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1 comment:

Peter Ash said...

I presented three alternate solutions of part (a). I've realized that one of the solutions had a mistake. I corrected the mistake and have added a fourth slightly different solution. See