Mathematical golf

Item 43 in Clement Wood’s 74-item Book of Mathematical Oddities (1927) is titled “A Matter of Golf”:

The nine holes of a mathematical golf course are 300, 250, 200, 325, 275, 350, 225, 375, and 400 yards apart. If a man could always strike the ball in a perfectly straight line exactly one of two distances, so that it could either go toward the hole, or pass over it, or drop into it, what would the two distances be that would carry him in the least number of strokes around the whole course? How many strokes would it require?

Solution: Choose a 125-yard driver and a 100-yard putter. Then you can cover the nine holes in 3, 2, 2, 3, 4, 3, 2, 3, and 4 strokes respectively, for a total of 26 strokes. (On the fifth hole, $$275 = 3\times 125 - 100$$.)

The general case here could be a pretty interesting mathematical recreation for long car trips. Let one person propose a “course” of several smallish numbers; for example 10, 14, 17, 28. Each player then quietly chooses their own “driver” and “putter” numbers. Then players tally their stroke count for each hole, and the low score wins.

For “10, 14, 17, 28” I think everyone would choose 10 and 7, covering the four holes in 1, 2, 2, and 4 strokes respectively. So that was an easy course. Now how about “10, 14, 18, 28”?

For “10, 14, 18, 28” I’d choose 10 and 8, covering the four holes in 1, 4, 2, and 3 strokes respectively, for a score of 10. But the best choice for this course is actually 14 and 4, which scores 2, 1, 2, 2 respectively for a score of 7 strokes!

Finally, consider the course “10, 15, 16, 29.” I’d stare perplexedly out over the course for several minutes before shrugging and choosing a 5 driver and a 4 putter (for a total of 16 strokes). But in fact you can do much better than I did! Can you finish this course in only 11 strokes?

If the basic game is too simple, you might consider adding the rule that the ball will still drop into the hole even if you overshoot by 1; but of course not if you undershoot by 1.

Now the best score for “10, 14, 17, 28” drops from 9 to 7. And “10, 15, 16, 29” can now be done in 8 strokes! Do you see how?

Posted 2023-03-23