2012 UNCO Math Contest II Problems/Problem 10

Problem

An integer equiangular hexagon is a six-sided polygon whose side lengths are all integers and whose internal angles all measure $120^{\circ}$ .

(a) How many distinct (i.e., non-congruent) integer equiangular hexagons have no side length greater than $6$ ? Two such hexagons are shown.

$[asy] draw((0,0)--(1,0)--(4,3*sqrt(3))--(3,4*sqrt(3))--(-1,4*sqrt(3))--(-1-3*sqrt(3)/2,4*sqrt(3)-1.5)--cycle,black); MP("1",(.5,0),S);MP("6",(2.5,1.5*sqrt(3)),SE);MP("2",(3.5,3.5*sqrt(3)),NE);MP("4",(1,4*sqrt(3)),N);MP("3",(-1-.75*sqrt(3),4*sqrt(3)-.75),NW);MP("5",(-.5-.75*sqrt(3),2*sqrt(3)-.75),W); draw((8,0)--(11,0)--(13,2*sqrt(3))--(11.5,3.5*sqrt(3))--(7.5,3.5*sqrt(3))--(6,2*sqrt(3))--cycle,black); MP("3",(9.5,0),S);MP("4",(12,sqrt(3)),SE);MP("3",(12.25,2.75*sqrt(3)),NE);MP("4",(9.5,3.5*sqrt(3)),N);MP("3",(6.75,2.75*sqrt(3)),NW);MP("4",(7,sqrt(3)),W); [/asy]$

(b) How many distinct integer equiangular hexagons have no side greater than $n$ ? Give a closed formula in terms of $n$ .

(A figure and its mirror image are congruent and are not considered distinct. Translations and rotations of one another are also congruent and not distinct.)

Solution

(a) $126$ (b) $\binom{n+3}{4}$

2012 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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2012 UNCO Math Contest II Problems/Problem 10

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