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

From ProofWiki
Jump to navigation Jump to search

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

The Tube Stairs
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.


Solution

$839$ steps.


Proof

Let $n$ be the number of steps in the tube station.

Let $1$ be added to $n$ to make $m$.

We have that:

$m$ is divisible by $2$
$m$ is divisible by $3$

and so on, until:

$m$ is divisible by $7$

Thus $m$ is divisible by the lowest common multiple of $\set {2, 3, 4, 5, 6, 7}$.

Hence we calculate:

\(\ds \lcm \set {2, 3, 4, 5, 6, 7}\) \(=\) \(\ds \lcm \set {2, 3, 2^2, 5, 2 \times 3, 7}\)
\(\ds \) \(=\) \(\ds \lcm \set {2^2, 3, 5, 7}\)
\(\ds \) \(=\) \(\ds 4 \times 3 \times 5 \times 7\)
\(\ds \) \(=\) \(\ds 420\)

and we note in passing that $420 = 21 \times 20$ and so is also divisible by $20$

As we are told there are nearly $1000$ steps, it is clear that $420$ is too small for $m$, so we multiply it by $2$ to get $840$.

We then note that $3 \times 420$ is way over $1000$ so cannot be the value for $m$.

So the only possible value for $m$ is indeed $840$.

Hence:

$n = 840 - 1 = 839$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Puzzles_and_Curious_Problems/73_-_The_Tube_Stairs/Solution&oldid=592434"