My Math Forum Polyhedra problem

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 June 19th, 2011, 05:53 AM #1 Member   Joined: Dec 2009 Posts: 65 Thanks: 0 Polyhedra problem Hello! I don't understand the solution of this problem. Please, help me. Problem: Consider a polyhedra with 9 vertices, all of them having integer coordinates. Prove that there exists an other lattice point in the interior of the polyhedra. Solution: The points $(x_1, x_2, x_3)$ and $(y_1, y_2, y_3)$ are considered to belong to the same set, if $x_i - y_i$ is even, $\forall i= 1,2,3$. In this way the set of lattice points is partitioned to 8 sets (??? why 8, and what are these sets ???), so there exists at least two points in the same set (it is clear, we use the pigeonhole principle). The midpoint of this segment determined by these points is also a lattice point because the coordinates of the midpoint are $\left( \frac{x_1 + y_1}{2}, \frac{x_2 + y_2}{2}, \frac{x_3 + y_3}{2} \right)$. Please help, many thanks, Crouch.
 June 19th, 2011, 11:33 AM #2 Senior Member     Joined: Feb 2010 Posts: 706 Thanks: 141 Re: Polyhedra problem The eight groups are based on the parity of the coordinates. E = even number, O = odd number. (E,E,E), (E,E,O), (E,O,E), (O,E,E) (O,O,O), (O,O,E), (O,E,O), (E,O,O) Since (E+E)/2 and (O+O)/2 are both integers, if two points lie in the same group then their midpoint must have integer values.

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