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

by : $272$

 * A Madam Problem

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 $80$.