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Nine distinct points with all coordinates integral are selected in the space. Prove that the line segment with ends at certain two of these points contains in its interior a point with all coordinates integral.

Jebediah
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1 Answers1

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HINT: Look at the coordinates modulo $2$ and apply the pigeonhole principle. The point in the interior of the segment will actually be its midpoint.

Brian M. Scott
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  • Can you elaborate more on this? – Jebediah Nov 02 '13 at 15:50
  • 9>2^3, so there must be 2 points in a set with identical coordinates modulo 2. – nsg Nov 02 '13 at 20:20
  • @Jebediah: If two integers are both odd or both even, their sum is divisible by $2$, so their midpoint (average) is an integer. You want two points whose coordinates match in parity (oddness/evenness). See also nsg’s comment. – Brian M. Scott Nov 03 '13 at 06:05