Henry Ernest Dudeney/Modern Puzzles/179 - Two Eight-Pointed Stars/Solution
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Modern Puzzles by Henry Ernest Dudeney: $179$
- Two Eight-Pointed Stars
- 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.
- 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.
Solution
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 $8$.
Historical Note
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.
Sources
- 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