Henry Ernest Dudeney/Puzzles and Curious Problems/272 - A Madam Problem/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $272$

A Madam Problem

In how many different ways is it possible to read the word $\text {MADAM}$ in the diagram?

You may go as you please, upwards and downwards, forwards and backwards,

any way possible along the open paths.

But the letters in every case must be contiguous, and you may never pass a letter without using it.

Solution

$80$.

Proof

There are $4$ instances of $\text M$, and each is symmetrically disposed.

Hence it is sufficient to count the number of ways you can form $\text {MADAM}$ from one of these and multiply by $4$.

So to proceed.

Starting with any given $\text M$, you can go:

$1$: Straight down the spoke of the wheel to the $\text M$ opposite

$2$: Around the edge clockwise to the next $\text M$

$3$: Around the edge anticlockwise to the next $\text M$

$4$: Down the spoke to the $\text D$ and back again to the $\text M$ you started at

$5$: Around the edge clockwise to the $\text D$ and back again to the $\text M$ you started at

$6$: Around the edge anticlockwise to the $\text D$ and back again to the $\text M$ you started at

$7$: Down the spoke of the wheel to the $\text D$ and then up the spoke to the left to the $\text M$

$8$: Down the spoke of the wheel to the $\text D$ and then up the spoke to the right to the $\text M$

$9$: Around the edge clockwise to the $\text D$, then along the diagonal via the $\text A$, to the adjacent $\text M$

$10$: Around the edge clockwise to the $\text D$, then along the diagonal to the $\text A$ back to the starting $\text M$

$11$: Around the edge anticlockwise to the $\text D$, then along the diagonal via the $\text A$, to the adjacent $\text M$

$12$: Around the edge anticlockwise to the $\text D$, then along the diagonal to the $\text A$ back to the starting $\text M$

$13$: Down the spoke to the $\text A$, then back up to the clockwise $\text D$ on the rim, and around the rim to the adjacent $\text M$

$14$: Down the spoke to the $\text A$, then back up to the clockwise $\text D$ on the rim, down the diagonal to the next $\text A$, and then up to the adjacent $\text M$

$15$: Down the spoke to the $\text A$, then back up to the anticlockwise $\text D$ on the rim, and around the rim to the adjacent $\text M$

$16$: Down the spoke to the $\text A$, then back up to the anticlockwise $\text D$ on the rim, down the diagonal to the next $\text A$, and then up to the adjacent $\text M$

$17$: Down the spoke to the $\text A$, then back up to the clockwise $\text D$ on the rim, then back around the rim to the $\text M$ you started at

$18$: Down the spoke to the $\text A$, then back up to the clockwise $\text D$ on the rim, then back the way you came to the $\text M$ you started at

$19$: Down the spoke to the $\text A$, then back up to the anticlockwise $\text D$ on the rim, then back around the rim to the $\text M$ you started at

$20$: Down the spoke to the $\text A$, then back up to the anticlockwise $\text D$ on the rim, then back the way you came to the $\text M$ you started at

That is $20$.

Hence the total number of ways is $80$.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $272$. -- A Madam Problem
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $429$. A Madam Problem