Henry Ernest Dudeney/Puzzles and Curious Problems/73 - The Tube Stairs

Puzzles and Curious Problems by Henry Ernest Dudeney: $73$

We ran up against Percy Longman, a young athlete, the other day when leaving Curley Street tube station.

He stopped at the lift, saying, "I always go up by the stairs.

A bit of exercise, you know.

But this is the longest stairway on the line -- nearly $1000$ steps.

I will tell you a queer thing about it that only applies to one other smaller stairway on the line.

If I go up two steps at a time, there is one step left for the last bound;

if I go up three at a time, there are two steps left;

if I go up four at a time, there are three steps left;

five at a time, four are left;

six at a time, five are left;

and if I went up seven at a time there would be six risers left over for the last bound.

Now, why is that?"

As he went flying up the stairs, three steps at a time, we laughed and said,

He little suspects that if he went up twenty steps at a time there would be nineteen risers for his last bound!"

How many risers are there in the Curley Street tube stairway?

The platform does not count as a riser, and the top landing does.

1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Locomotion and Speed Puzzles: $73$. -- The Tube Stairs
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Speed & Distance Puzzles: $87$. The Subway Stairs