Lattice Points II

Probability Level 5

A lattice point is defined as a point in the two-dimensional plane with integral coordinates. We define the centroid of three points ( x i , y i ) , i = 1 , 2 , 3 (x_i,y_i), i=1,2,3 as point ( x 1 + x 2 + x 3 3 , y 1 + y 2 + y 3 3 ) . \left(\frac{x_1+x_2+x_3}3,\frac{y_1+y_2+y_3}3\right).

What is the largest number of distinct lattice points in the plane such that the centroid of any three of them is not a lattice point?


The answer is 8.

Brian Lie by Brian Lie , 26, China

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1 solution

Mark Hennings
May 23, 2019

The Kemnitz (no longer a) Conjecture states that, for any positive integer n n , whenever we have 4 n − 3 4n-3 lattice points, there is a collection of n n if them whose centroid is a lattice point. Thus, in any set of 9 9 lattice points, we can always find a set of 3 3 whose centroid is a lattice point. It is easy to find 8 8 lattice points, no three of which have a lattice point centroid, for example (0,0), (3,0), (1,0), (1,3), (0,1), (3,1), (1,1), (4,4).

Thus the answer is 8 \boxed{8} .

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