Henry Ernest Dudeney/Modern Puzzles/123 - Making an Octagon/Solution

by : $123$

 * Making an Octagon
 * ''Can you cut the regular octagon from a square piece of paper without using compasses or ruler,
 * or anything but scissors?
 * You can fold the paper to make creases.


 * Dudeney-Modern-Puzzles-123.png

Solution
Let the square be $\Box ABCD$.


 * Dudeney-Modern-Puzzles-123-solution.png

Fold $AD$ to $BC$ and $AB$ to $DC$ to obtain the midpoints $E$, $F$, $G$ and $H$ of $AD$, $BC$, $AB$ and $DC$ respectively.

Fold $EG$, $FG$, $EH$ and $FH$.

Fold at $H$ to place $DH$ against $EH$, then at $E$ to place $ED$ against $EH$.

Where those folds cross at $I$ is one of the vertices of the regular octagon we are seeking.

The other three vertices are obtained similarly.