# Sunday Times Teaser 2928 – Golf Balls

### by John Owen

#### Published November 4 2018 (link)

Three friends play 18 holes of golf together every week. They have a large supply of golf balls numbered from 1 to 5. At the beginning of a year, they each start playing with a new ball, but all three ball numbers are different. Alf uses the same ball number for exactly 21 holes before changing to the next higher number (or to number 1 if he was using a number 5). He continues to use each number for exactly 21 holes. The same applies to Bert, except that he changes his ball number in the same way after every 22 holes, and to Charlie who changes his ball number after every 23 holes.

Last year, there was just one occasion when they all used the same ball number on a hole.

What was the number of the hole on that occasion?

1. Erling Torkildsen has developed an interactive GeoGebra model for this teaser here.

2. To make modular arithmetic easier, I modified the puzzle so that the balls are numbered 0,1,2,3,4, holes are 0 to 17 and weeks are 0 to 52, then with starting balls a,b,c
set h = 18*week + hole.,
The balls coincide when
(a + (h – h mod 21)/21) mod 5 = (b + (h – h mod 22)/22) mod 5 =(c + (h – h mod23)/23) mod 5

Then h can be converted back to a hole in the range 1 to 18

I didn’t spot that one of the balls could be fixed until I saw Brian’s solution, but using that observation:

3. 