My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum

LinkBack Thread Tools Display Modes
June 19th, 2011, 05:53 AM   #1
Joined: Dec 2009

Posts: 65
Thanks: 0

Polyhedra problem


I don't understand the solution of this problem. Please, help me.

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.

The points and are considered to belong to the same set, if is even, . 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

Please help, many thanks,
Crouch is offline  
June 19th, 2011, 11:33 AM   #2
Senior Member
mrtwhs's Avatar
Joined: Feb 2010

Posts: 711
Thanks: 147

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.
mrtwhs is offline  

  My Math Forum > High School Math Forum > Algebra

polyhedra, problem

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
How many differently polyhedra can be obtained? mathLover Algebra 4 April 17th, 2012 02:02 AM
Question about polyhedra without diagonals Maurice Applied Math 0 January 30th, 2012 09:42 AM
Polyhedra Algebra 7 November 21st, 2010 04:59 AM

Copyright © 2019 My Math Forum. All rights reserved.