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

Modern Puzzles by Henry Ernest Dudeney: $30$

This is a rough sketch of a race up a staircase in which $3$ men took part.

Ackworth, who is leading, went up $3$ risers at a time, as arranged;

Barnden, the second man, went $4$ risers at a time,

and Croft, who is last, went $5$ at a time.

Undoubtedly Ackworth wins.

But the point is,

How many risers are there in the stairs, counting the top landing as a riser?

There are $19$ risers.

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.

$\blacksquare$