**Solution**

The scores must have been 10, 15, 9, 20, 12. This solution depends on the fact that, by the rules of the game, certain scores (7, 11, 13, 14, 17, etc.) can not occur because they can not be formed by multiplying together two numbers between 1 and 6.

There are eight possibilities for the first two throws:

1 & 6

3 & 8

4 & 9

5 & 10

10 & 15

15 & 20

20 & 25

25 & 30

But only five of these eight give a valid score for the third throw, 6 less than the second. Of these five, there are only two which give a valid forth score 11 greater than the third, and only one which which gives a valid fifth score 8 less than the fourth.