Henry Ernest Dudeney/Modern Puzzles/177 - The Six-Pointed Star

by : $177$

 * The Six-Pointed Star
 * We have considered the question of the five-pointed star.
 * We shall now find the six-pointed star even more interesting.


 * In this case we can always use the twelve consecutive numbers $1$ to $12$ and the sum of the four numbers in every line will always be $26$.


 * The numbers at the six points of the star may add up to any even number from $24$ to $54$ inclusive, except $28$ and $50$, which are impossible.
 * It will be seen in the example that the six points add up to $24$.


 * Dudeney-Modern-Puzzles-177.png


 * If for every number in its present position you substitute its difference from $13$ you will get another solution, its complementary,
 * with the points adding up to $54$, which is $78$ less $24$.
 * The two complementary totals will always sum to $78$.


 * I will give the total number of different solutions and point out some of the pretty laws which govern the problem,
 * but I will leave the reader this puzzle to solve.
 * There are six arrangements, and six only, in which all the lines of four and the six points also add up to $26$.
 * Can you find one or all of them?