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

by : $30$

 * The Staircase Race
 * 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.


 * Dudeney-Modern-Puzzles-30.png


 * Undoubtedly Ackworth wins.
 * But the point is,
 * How many risers are there in the stairs, counting the top landing as a riser?


 * I have shown only the top of the stairs.
 * There may be scores, or hundreds, of risers below the line.
 * It was not necessary to draw them, as I only wanted to show the finish.
 * But it is possible to tell from the evidence the fewest possible risers in that staircase.


 * Can you do it?

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:
 * $N \equiv 1 \pmod 3$
 * $N \equiv 3 \pmod 4$
 * $N \equiv 4 \pmod 5$

The smallest such $N$ is $19$.