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Problem

In order to win a game of darts, a player must finish on exactly zero and their last dart must land in a double or hit the bull's eye.

For example, if a player hit treble twenty (60 points), double twenty (40 points) and the bull's eye (50 points) they would score 150 points and could use this combination to finish.

How many ways can a player finish from 150 points with three darts?

Solution

If we finish on a bull's eye: 150 50 = 100, we must make 100 with the first two darts: 40,60,50 / 50,50,50 / 60,40,50

The highest double to finish on is 2 20 = 40 and 150 40 = 110; which can be made in two ways: 50,60,40 / 60,50,40

The highest score possible with first two darts is 2 60 = 120, so the lowest double is 150 120 = 30, i.e. double 15.

With the exception of finishing on a bull's eye the first two darts must score over 100, which means they must both be trebles and so the sum of their scores will be divisible by 3. As 150 is divisible by 3, the double must also be divisible by 3.

Double 15, 150 30 = 120: 60,60,30
Double 18, 150 36 = 114: 54,60,36 / 60,54,36 / 57,57,36

And so there are 9 ways of finishing on 150.

What is the highest score it is possible to finish on in a game of darts?

Investigate the number of ways of finishing on different scores.