January 2024
Problem of the Month

Golf with Tiger Woods
by Kipp Johnson of Valley Catholic School, Beaverton, Oregon

Last weekend I played eighteen holes of golf with Tiger Woods.
We bet $1 on the first hole, and doubled the bet on each succeeding hole.
He got lucky and beat me on the first seventeen holes, but I won the last hole.
(There were no ties on any hole.).

How much did I win or lose?

Solution:

The answer is that I won one dollar.

The formula for the sum of a finite geometric progression tells us that 2⁰ + 2¹ + 2² + ... 2^(n-1) = 2ⁿ -1.

In other words, if we start with 2⁰ = 1, the sum of any number of consecutive terms of the geometric progression with common ratio 2 always equals one less than the next power of 2.   Thus after being deeply in debt, I wound up winning $1.

(This problem illustrates the Martingale Betting System.   Wikipedia has a good discussion of it.)

Here is the formula for the sum of a geometric series with the first term a, common ratio r, and n number of terms:


So, Tiger Woods' winnings would be:


My winnings would be:
My winnings = 2¹⁷ = $131,072

Correctly solved by:

1. K. Sengupta	Calcutta, India
2. Kamal Lohia	Holy Angel School, Hisar, Haryana, India
3. Davit Banana	Istanbul, Turkey
4. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
5. Seth Cohen	Concord, New Hampshire, USA
6. Ivy Joseph	Pune, Maharashtra, India
7. Heather Widener	McLean, Virginia, USA
8. Colin (Yowie) Bowey	Beechworth, Victoria, Australia
9. Kelly Stubblefield	Mobile, Alabama, USA

Send any comments or questions to: David Pleacher