This puzzle is about golf, but does not require any specific golf knowledge. Rounds of golf are played over 18 holes (the front nine and the back nine), and each hole can be won by either player or it can be drawn (or halved in golfing parlance). However this puzzle would work equally well with two lots of 18 games of chess, for example, as games of chess can be won/lost or drawn.
A father organised a round of golf with his two sons, and to try to raise their interest in the sport he proposed the following inducement:
- I will play the first nine holes against Andy and the second nine holes against Ben.
- If I don't win the first hole I'll pay Andy £1 and if I don't win the tenth hole I'll pay Ben £1.
- If I don't win a hole immediately after winning a hole I'll also pay £1.
- If I don't win a second consecutive hole I'll pay £2 for that hole, £3 for a third consecutive hole, and so on.
In short, to win money each son has to prevent his father winning a hole, and the amount won on a hole increases the more consecutive holes the father fails to win.
The father had a disastrous round winning only five holes; however, he did win the first and the 18th - the two holes in sight of the clubhouse, which saved a little embarrassment.
And his two sons won the same amount of money. So how much money did the father pay out in total?
We know that the father won only five holes and won the first (against Andy) and the 18th (against Ben). So he must have won three other holes, and we can see that he must have won one of these against one son and two of these against the other son. Otherwise one son would win much more money than the other.
Examining the possible sequences for the nine holes when the father won two holes:
- A losing sequence of seven holes would cost £28
- Losing sequences of six holes and one hole would cost £21 plus £1 totalling £22
- Losing sequences of five holes and two holes would cost £15 plus £3 totalling £18
- Losing sequences of four holes and three holes would cost £10 plus £6 totalling £16
Examining the possible sequences for the nine holes when the father won three holes:
- A losing sequence of six holes would cost £21
- Losing sequences of five holes and one hole would cost £15 plus £1 totalling £16
- Losing sequences of four holes and two holes would cost £10 plus £3 totalling £13
- Losing sequences of three holes and three holes would cost £6 plus £6 totalling £12
- Losing sequences of four holes, one hole and one hole would cost £10 plus £1 plus £1 totalling £12
- Losing sequences of three holes, two holes and one hole would cost £6 plus £3 plus £1 totalling £10
- Losing sequences of two holes, two holes and two holes would cost £3 plus £3 plus £3 totalling £9
These are the only possible losses to the two sons and we know that they won the same amount of money. The only total in both lists is £16 and so each son must have won this amount, and thus the father paid out £32.