Henry Ernest Dudeney/Modern Puzzles/170 - A Heptagon Puzzle/Solution

by : $170$

 * A Heptagon Puzzle

Solution

 * The diagram shows the solution.


 * Dudeney-Modern-Puzzles-170-solution.png


 * Starting at the highest point, write the numbers $1$ to $7$ in a clockwise direction at alternate points.
 * Then, starting just above the $7$, write $8$ to $14$ successively in the opposite direction, taking every vacant circle in turn.


 * If instead you write in $1$, $3$, $5$, $7$, $9$, $11$, $13$ and then $2$, $4$, $6$, $8$, $10$, $12$, $14$,
 * you will get a solution with the sides adding to $22$ instead of $19$.


 * If you substitute for every number in these solutions its difference from $15$ you will get the complementary solutions,
 * adding respectively to $26$ and $23$ (the difference of $19$ and $22$ from $45$).