Prove a circle contains at least seven points in a regular hexagon point-placement problem

Key takeaway: This is one of those geometry problems where the shape suggests a natural partition. Once you cut the hexagon into equal regions, the pigeonhole principle does most of the heavy lifting.

We are given 37 points in a regular hexagon of side 1, and we want to prove that at least 7 of them lie in some circle of radius $\frac{\sqrt3}{3}$.

1) Partition the hexagon

Join the center of the regular hexagon to its six vertices. This divides the hexagon into 6 congruent equilateral triangles, each of side 1.

The area is not the key point here; the important part is that each small triangle has diameter 1. In an equilateral triangle of side 1, the maximum distance between two points is 1.

2) Put the points into the 6 triangles

With 37 points distributed among 6 triangles, the pigeonhole principle gives at least

$$\left\lceil\frac{37}{6}\right\rceil=7$$

points in one triangle.

3) Fit that triangle inside a circle of radius $\frac{\sqrt3}{3}$

Each equilateral triangle of side 1 has circumradius

$$R=\frac{1}{\sqrt3}=\frac{\sqrt3}{3}.$$

So the triangle is contained in a circle of radius $\frac{\sqrt3}{3}$.

Therefore, the 7 points lying in that triangle also lie within that circle.

Hence at least 7 of the points are contained in some circle of radius $\frac{\sqrt3}{3}$.

Why this works

The geometry gives a natural partition, and the pigeonhole principle forces one region to contain enough points. After that, the circumcircle bound finishes the job cleanly.

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Pitfalls the pros know 👇 Two common slips: first, using the inradius instead of the circumradius of the equilateral triangle; second, forgetting that the circle must contain the whole triangle, not just its center. The correct radius here is $1/\sqrt3$, not $1/(2\sqrt3)$ or $1/2$.

What if the problem changes? If the number of points were 31, the same partition would only guarantee 6 points in one triangle, since $\lceil 31/6\rceil=6$. So the threshold for forcing 7 points is exactly when the total exceeds 36.

Tags: pigeonhole principle, equilateral triangle, circumradius