This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

Divide the hexagon into:

1) 3 identical rhombuses.

2) 6 identical kites(?).

3) 4 identical trapezoids (trapeziums in Britain).

4) 8 identical shapes (any shape).

5) 12 identical shapes (any shape).

geometry/dissections/hexagon.s

What is considered 'identical' for these questions? If mirror-image shapes

are allowed, these are all pretty trivial. If not, the problems are rather

more difficult...

1. Connect the center to every second vertex.

2. Connect the center to the midpoint of each side.

3. This is the hard one. If you allow mirror images, it's trivial:

bisect the hexagon from vertex to vertex, then bisect with a

perpendicular to that, from midpoint of side to midpoint of side.

4. This one's neat. Let the side length of the hexagon be 2 (WLOG).

We can easily partition the hexagon into equilateral triangles

with side 2 (6 of them), which can in turn be quartered into

equilateral triangles with side 1. Thus, our original hexagon

is partitioned into 24 unit equilateral triangles. Take the

trapezoid formed by 3 of these little triangles. Place one such

trapezoid on the inside of each face of the original hexagon, so

that the long side of the trapezoid coincides with the side of the

hexagon. This uses 6 trapezoids, and leaves a unit hexagon in the

center as yet uncovered. Cover this little hexagon with two of

the trapezoids. Voila. An 8-identical-trapezoid partition.

5. Easy. Do the rhombus partition in #1. Quarter each rhombus by

connecting midpoints of opposite sides. This produces 12 small

rhombi, each of which is equivalent to two adjacent small triangles

as in #4.

Except for #3, all of these partitions can be achieved by breaking up the

hexagon into unit equilateral triangles, and then building these into the

shapes desired. For #3, though, this would require (since there are 24 small

triangles) trapezoids formed from 6 triangles each. The only trapezoid that

can be built from 6 identical triangles is a parallelogram; I assume that the

poster wouldn't have asked for a trapezoid if you could do it with a special

case of trapezoid. At any rate, that parallelogram doesn't work.

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