Henry Ernest Dudeney/Puzzles and Curious Problems/283 - Pat in Africa/Solution

by : $283$

 * Pat in Africa

Solution
Pat was expected to start at $9$ and count $29$.

This would have counted out all the natives.

However, instead he started the count at $1$ (himself) and counted $11$.

Proof
We are given the order of the men:
 * $W_1, N_2, W_3, N_4, N_5, W_6, N_7, W_8, N_9, W_{10}$

Counting $11$ from $W_1$ (which is Pat) takes out $W_1$ himself, leaving:
 * $N_2, W_3, N_4, N_5, W_6, N_7, W_8, N_9, W_{10}$

Counting $11$ from $N_2$ takes out $W_3$, leaving:
 * $N_2, N_4, N_5, W_6, N_7, W_8, N_9, W_{10}$

Counting $11$ from $N_4$ takes out $W_6$, leaving:
 * $N_2, N_4, N_5, N_7, W_8, N_9, W_{10}$

Counting $11$ from $N_7$ takes out $W_{10}$, leaving:
 * $N_2, N_4, N_5, N_7, W_8, N_9$

Counting $11$ from $N_2$ takes out $W_8$, leaving:
 * $N_2, N_4, N_5, N_7, N_9$

So all Westerners are taken out for a flogging, while the natives escape.

Let us count $29$ starting at $N_9$.

Counting $29$ from $N_9$ takes out $N_7$, leaving:
 * $W_1, N_2, W_3, N_4, N_5, W_6, W_8, N_9, W_{10}$

Counting $29$ from $W_8$ takes out $N_9$, leaving:
 * $W_1, N_2, W_3, N_4, N_5, W_6, W_8, W_{10}$

Counting $29$ from $W_{10}$ takes out $N_4$, leaving:
 * $W_1, N_2, W_3, N_5, W_6, W_8, W_{10}$

Counting $29$ from $N_5$ takes out $N_5$, leaving:
 * $W_1, N_2, W_3, W_6, W_8, W_{10}$

Counting $29$ from $W_6$ takes out $N_2$, leaving:
 * $W_1, W_3, W_6, W_8, W_{10}$

So in this case, all the natives are flogged and the Westerners escape.