Henry Ernest Dudeney/Modern Puzzles/108 - Hexagon to Square/Solution

Modern Puzzles by Henry Ernest Dudeney: $108$

Can you cut a regular hexagon into $5$ pieces that will fit together to form a square?

Cut the hexagon in half along a diagonal and place the two parts together to form the parallelogram $ABCD$.

Construct $CF$ perpendicular to $DC$.

Produce $DC$ to $E$ where $CE = CF$.

Construct the semicircle $DHE$, whose center will be at $G$.

Produce $CF$ to $H$.

We have that $CH^2 = DC \cdot CE$.

We have that $DC \cdot CE$ is the area of the parallelogram $ABCD$, and therefore the original hexagon.

Hence a square whose side is $CH$ will have the same area as that hexagon.

Construct the semicircle $DJC$, whose center will be at $K$.

Construct the arc $HJ$ whose center is at $C$.

Draw $CJ$ and $DJ$.

Construct $LJ = CJ$ and the follows.

The rest explains itself.

$\blacksquare$

Henry Ernest Dudeney claims that he was the first to publish this construction, citing the Weekly Dispatch for August $1901$.

However, Martin Gardner reports that Victor Meally unearthed the information that in fact it had previously been discovered by Paul Busschop.

This was published in Volume $\text {II}$ of Nouvelle correspondance mathématique in $1875$, on page $83$.