Problem: Let there be 9 points in 3-space with integer coordinates. Show that th
ID: 1893058 • Letter: P
Question
Problem:Let there be 9 points in 3-space with integer coordinates.
Show that there is a pair of these points whose line
segment contains an interior point whose coordinates are
integers.
Outline of Proof:
Points in 3-space have 3 coordinates, (a,b,c).
Integer coordinates are either odd or even.
There are 8 odd-even patterns of integer coordinates in 3-space.
Since there are 9 points, at least two must have the same pattern
by the Pigeon-Hole Principle.
The midpoint of the line segment joining two points with the
same pattern has integer coordinates.
How do you fill in this proof? What are the missing details? How do you know if you filled it in completely?
Explanation / Answer
1) "There are odd-even patterns of integer coordinates in 3-space" makes sense because each coordinate, of which there are three, can be either even or odd, so there are 23 = 8 patterns. Everything before this statement is intuitive and obvious.
2) "same pattern" should be clarified: there must exist, by the Pigeon-hole Principle, a pair of points (a,b,c) and (d,e,f) such that the three pairs a and d, b and e, c and f each have the same parity.
3) "The midpoint of the line segment joining two points with the same pattern has integer coordinates" needs more details. The midpoint of two points (a,b,c) and (d,e,f) is M = ((a+d)/2, (b+e)/2, (c+f)/2). Because the "same pattern" coordinates have the same parity, each of (a+d), (b+e), and (c+f) must be even, so M must have integer coordinates.
Thus our proof by construction took 9 arbitrary points, found a pair of points with coordinates of equal corresponding parity using the Pigeon-hole Principle, and found that the midpoint of these two special points has integer coordinates, so that the interior point of the line segment with integer coordinates is found to always exist.
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