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

Modern Puzzles by Henry Ernest Dudeney: $177$

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 $24$.

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?

1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Magic Star Problems: $177$. -- The Six-Pointed Star
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Combinatorial & Topological Problems: Magic Star Puzzles: $394$. The Six-Pointed Star