Henry Ernest Dudeney/Modern Puzzles/179 - Two Eight-Pointed Stars/Solution

Modern Puzzles by Henry Ernest Dudeney: $179$

The star may be formed in two different ways, as shown in our diagram, and the first example is a solution.

The numbers $1$ to $16$ are so placed that every straight line of four adds up to $34$.

If you substitute for every number its difference from $17$ you will get the complementary solution.

$\qquad$

Let the reader try to discover some of the other solutions, and he will find it a very hard nut, even with this one to help him.

But I will present the puzzle in an easy and entertaining form.

When you know how, every arrangement in the first star can be transferred to the second one automatically.

Every line of four numbers in the one case will appear in the other, only the order of the numbers will have to be changed.

Now, with this information given, it is not a difficult puzzle to find a solution for the second star.

Dudeney presents this solution:

If you find any solution to one of the stars, you can immediately transfer it to the other by noting the relative positions in the case given.

No attempt at this stage has been made to find out how many stars there are of order $8$.

In $1963$, A. Domergue analysed this problem and found $112$ solutions.

He was also able to estimate that the $9$-pointed star has more than $2000$ distinct patterns.

1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $179$. -- Two Eight-Pointed Stars
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $396$. Two Eight-Pointed Stars