Jigsaw Hexagon

Geometry Level 4

An alternative style of jigsaw puzzle has pieces in the shape of equilateral triangles with sides one unit long. When the puzzle is complete, it is a regular hexagon, with 6 sides of 15 units.

The puzzle contains $e$ edge pieces (one side of the triangle forms the outside of the puzzle) and $i$ interior pieces (all sides of the triangle are connected to other pieces.

How much are $e$ and $i$ ? Report the answer by concatenating the numerals; e.g. if $e = 32$ and $i = 177$ , type $32177$ .

The answer is 901260.

1 solution

Dan Ley
Feb 14, 2017

We can generalise the number ( $n$ ) of unit equilateral triangles that make up a larger equilateral triangle of length $a$ with $n=a^2$ .

So that means the number of unit triangles inside a larger triangle of length $a=15$ is simply $15^2=225$ , and as 6 of these triangles make up a hexagon, there are $6\times 225=1350$ unit triangles in total.

Clearly, each large triangle that makes up the hexagon contributes 15 edge pieces, giving $e=15\times 6=90$ . The interior pieces are the ones left over, i.e. $i=1350-90=1260$ .

So the answer is $\overline{ei}=901260$ .

Jigsaw Hexagon

The answer is 901260.

1 solution

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