This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three steps at a time, as arranged; Barnden, the second man, went four steps at a time, and Croft, who is last, went five at a time. Undoubtedly Ackworth wins. But the point is, how many steps are there on the stairs, counting the top landing as a step? The top of the stairs are only shown. There may be scores, or hundreds, of steps below the line. But it is possible to tell from the evidence the fewest possible steps in that staircase. Can you do it?
Given: Ackworth is short of reaching the top by only one step.
Puzzle – The Staircase Race
Solution: If the staircase were such that each man would reach the top in a certain number of full leaps, without taking a reduced number at his last leap, then the smallest possible number of steps would, of course, be 60 (that is, 3 X 4 X 5). But it is given to us that,
- Ackworth while taking three steps at a leap, has one odd step at the end,
- Barnden taking four at a leap will have three only at the end and,
- Croft, taking five at a leap, will have four only at the finish.
Therefore, we have to find the smallest number that, when divided by 3, leaves a remainder of 1, when divided by 4 leaves 3, and when divided by 5 leaves a remainder of 4. This number is 19. So there were 19 steps in all.