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

Source Work

1926: Henry Ernest Dudeney: Modern Puzzles:

Magic Star Problems

$177$ -- The Six-Pointed Star

1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems:

Combinatorial & Topological Problems

Magic Star Puzzles

$394$. The Six-Pointed Star

Mistake

There are $37$ solutions in all, or $74$ if we count complementaries.

$32$ of these are regular, and $5$ are irregular.

Of the $37$ solutions, $6$ have their points summing to $26$. These are as follows:

$\begin {array} {rrrrrrrrrrrr}

10 & 6 & 2 & 3 & 1 & 4 & 7 & 9 & 5 & 12 & 11 & 8 \\ 9 & 7 & 1 & 4 & 3 & 2 & 6 & 11 & 5 & 10 & 12 & 8 \\ 5 & 4 & 6 & 8 & 2 & 1 & 9 & 12 & 3 & 11 & 7 & 10 \\ 5 & 2 & 7 & 8 & 1 & 3 & 11 & 10 & 4 & 12 & 6 & 9 \\ 10 & 3 & 1 & 4 & 2 & 6 & 9 & 8 & 7 & 12 & 11 & 5 \\ 8 & 5 & 3 & 1 & 2 & 7 & 10 & 4 & 11 & 9 & 12 & 6 \\ \end {array}$

...

Also note that where the $6$ points add to $24$, $26$, $30$, $32$, $34$, $36$ or $38$, the respective number of solutions is $3$, $6$, $2$, $4$, $7$, $6$ and $9$, making $37$ in all.

Correction

There are in fact $80$ solutions, as noted by Martin Gardner in 536 Puzzles & Curious Problems.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $177$. -- The Six-Pointed Star
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $394$. The Six-Pointed Star
• 1975: Martin Gardner: Mathematical Carnival: $5$: Magic Stars and Polyhedrons