Question 1205940: A lattice point is a point with integer coordinates.
Find, with proof, the smallest 𝑁 such that, given any set of 𝑁 lattice
points, you can find a pair of them whose midpoint is also a lattice point.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
odd + odd = even
even + even = even
even + odd = odd
Adding two numbers of the same parity leads to an even result.
Any even number when divided by 2 will result in some integer.
If a and c have the same parity then (a+c)/2 is an integer.
If b and d have the same parity then (b+d)/2 is an integer.
If a and c differ in parity (one is odd, the other even) then (a+c)/2 isn't an integer.
A similar situation happens with (b+d)/2 as well.
Since we have 2 choices for parity and 2 coordinate slots, there are 2*2 = 4 different types of ordered pairs:
(even, even)
(even, odd)
(odd, even)
(odd, odd)
Let's say that we picked 4 random points and let's say we picked 1 of each form shown above. Clearly we don't have a parity match if we have this bad of luck. But the 5th point will guarantee to land on one of the parities mentioned due to the Pigeon-Hole Principle.
Therefore, we'll have a guaranteed parity match by the 5th point if there wasn't a match already.
In other words, having 5 random lattice points guarantees at least two of those points form a midpoint that's also a lattice point.
Answer: N = 5
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