Henry Ernest Dudeney/Modern Puzzles/178 - The Seven-Pointed Star/Solution

Modern Puzzles by Henry Ernest Dudeney: $178$

The Seven-Pointed Star

All you have to do is place the numbers $1$, $2$, $3$, up to $14$ in the fourteen discs so that every line of four disks shall add up to $30$.

Solution

Dudeney's solution is as follows:

Place $5$ in the top point.

Let the $4$ numbers in the horizontal line, that is $7$, $11$, $9$, $3$, be such that:

the two outside numbers sum to $10$

the inner numbers sum to $20$

the difference between the two outer numbers is twice the difference between the two inner numbers.

Then their complements with $15$ are placed in the relative positions indicated by the dotted lines.

The remaining $4$ numbers, that is $13$, $2$, $14$, $1$ are easily adjusted.

From this arrangement we can get $3$ others.

$(1): \quad$ Change the $13$ with the $1$ and the $14$ with the $2$.

$(2)$ and $(3): \quad$ Substitute in each of those two above arrangements its difference from $15$.

There are $56$ different arrangements, counting complements.

Class $\text I$ is those as above, where pairs in the positions $7 - 8$, $13 - 2$, $3 - 12$, $14 - 1$ all add to $15$, and there are $20$ such cases.

Class $\text {II}$ includes cases where pairs in the positions $7 - 2$, $8 - 13$, $3 - 1$, $12 - 14$ all add to $15$, and there are $20$ such cases.

Class $\text {III}$ includes cases where pairs in the positions $7 - 8$, $13 - 2$, $3 - 1$, $12 - 14$ all add to $15$, and there are $16$ such cases.

There are also another $16$ solutions which Dudeney did not find.

Historical Note

Martin Gardner reported this problem in his Mathematical Games column in Scientific American in the December $1965$ issue.

In correspondence, he received confirmation from Mrs. Peter W. Montgomery that there are in fact $72$ patterns for this seven-pointed star, rather than the $56$ offered by Dudeney.

This was confirmed by the independent work of E.J. Ulrich and A. Domergue.

In $1966$, Alan Moldon used a computer at the University of Waterloo in Ontario to confirm the $72$ solutions.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $178$. -- The Seven-Pointed Star
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $395$. The Seven-Pointed Star