Henry Ernest Dudeney/Modern Puzzles/30 - The Staircase Race/Solution

by : $30$

 * The Staircase Race

Solution
There are $19$ risers.

Proof
We refer to Ackworth, Barnden and Croft as $A$, $B$ and $C$.

Let $N$ be the number of risers.

The diagram shows that:
 * $A$ has $1$ odd step at the top
 * $B$ will have $3$ such odd steps
 * $C$ will have $4$ such steps.

Thus we have:
 * $N \equiv 1 \pmod 3$
 * $N \equiv 3 \pmod 4$
 * $N \equiv 4 \pmod 5$

Notice that:
 * $N + 1 \equiv 0 \pmod 4$
 * $N + 1 \equiv 0 \pmod 5$

so $N + 1$ must be a multiple of $20$.

The smallest such $N$ is $19$, and we see that it satisfies the first condition as well.