brainbuster no. 24 - Guided Bus

This competition closed on Sunday 25 November 2007.

the brainbuster

"The Melchester Guided Bus system has become increasingly popular following the introduction of a controversial congestion charge scheme.

The local council has therefore decided to add further stations on the guided bus route.

Every station on the guided bus route sells tickets to every other station.

When more stations are added, 46 sets of additional tickets will be required.

How many stations will be added to the guided bus system - and how many stations are there at present?"

the solution

Here is the answer as provided by our winner:

There are currently 11 stations and 2 stations are to be added.

The working:

In a system as described containing N stations, the number of different single tickets is:

(Eqn 1): N(N-1) = N^2 - N

If M stations are added then the new number of tickets will be:

(Eqn 2): (N+M)(N+M-1) = N^2 + NM - N + MN + M^2 - M

The difference will therefore be:

(Eqn 2) - (Eqn 1) = M^2 + M(2N-1)

Which we are told is equal to 46 giving us

(Eqn 3): M^2 + (2N-1)M -46 = 0

Using the quadratic formula, to solve for M, the discriminant is (2N-1)^2 + 184.

We know from the problem that M and N must be positive integers, so the discriminant must be a perfect square.

From the fact that the discriminant must be positive, we can deduce that N must be greater than 6.

Substituting N=7 into (Eqn 3) and solving gives M = 2.89.

This is an upper bound on M as can be seen by examination of (Eqn 3). This suggests two possible solutions for M: 1,2.

Substituting M = 2 into (Eqn 3) gives N = 11. Substituting M = 1 into (Eqn 3) gives N = 23.

The problem does state that "stations" (plural) are added, so the actual solution is that there are currently 11 stations and 2 stations are to be added.

and the lucky winner is...

Congratulations to Damian Smith, a Physicist and Firmware Engineer, who wins £200 of Firebox vouchers.

Many thanks to all entrants - do try brainbuster no. 25!

thinkingpeoplefirst