snookerA Maths Snooker Puzzle

This was suggested by Paul our Physics teacher – and is a nice little maths puzzle.

The maximum break score in snooker is 147 which is achieved by:
15 reds (1 point each) , 15 blacks (7 points each), then yellow ( 2 points), then green (3 points), brown (4 points), blue (5 points), pink (6 points) and finally black (7 points).

Now, if you wanted the maximum break in snooker to be 180, but wanted all the balls to still have a distinct value and for black to remain the highest ball, how could you change the values of the balls to get a 180 top score? And can you prove there is only one answer?

Answer below in white text (highlight to reveal)

1) Firstly we can show that if red has to be 1.  For red-black combinations (2,8) (2,9) we can show that no solution is possible.  For (2,10) and above, and for (3,9) and above we reach 180 without the other colours.
2) The only options are (1,8) (1,9) or (1,10).  (1,8) leaves us needing 37 from 5 balls valued between 2-7 – which we can’t do.  (1,10) leaves us needing 5 from 5 balls – which we can’t do.  (1,9) is the only possible solution – and this requires 21 from 5 balls valued (2,3,4,5,6,7,8).  This can only be achieved with 2,3,4,5,7.

Like puzzles?  Then you might also enjoy some other brain teaser posts here.