Question

In an equiangular hexagon, all interior angles measure $120^\circ.$ An example of such a hexagon with side lengths of $2,$ $3,$ $1,$ $3,$ $2,$ and $2$ is shown below, inscribed in equilateral triangle $ABC.$ Consider all equiangular hexagons with positive integer side lengths that can be inscribed in $\triangle ABC,$ with all six vertices on the sides of the triangle. What is the total number of such hexagons? Hexagons that differ only by a rotation or a reflection are considered the same.

$4$

$5$

$6$

$7$

$8$

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Written solution:

The example shows that each side of $\triangle ABC$ has length $6$ . Let $x,y,z$ be the small corner lengths cut off at $A,B,C$ , respectively. Then the other three side lengths of the hexagon are $6-x-y$ , $6-y-z$ , and $6-z-x$ . All six side lengths are positive integers exactly when $x,y,z$ are positive integers and each pair sum is less than $6$ . Up to rotation and reflection, this means counting unordered triples $x\le y\le z$ with $y+z\lt6$ : $(1,1,1)$ , $(1,1,2)$ , $(1,1,3)$ , $(1,1,4)$ , $(1,2,2)$ , $(1,2,3)$ , $(2,2,2)$ , and $(2,2,3)$ . There are $8$ such hexagons.

2026 AMC 8 Problem 25

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Problem 25 in Other Years