By Josh Feldman

A few months ago, I finally convinced my parents to get cable television for the first time. While I thought it was great that my folks finally had access to 200 channels instead of just eight, I admit I was nervous that my dad would just watch political news shows all day during his retirement. Surprisingly, my dad has kept his news watching to a minimum. However, he has found a new favorite network to watch: The Tennis Channel. Over on the clay court circuit, my dad has watched an impressive amount of tennis. While I knew he enjoyed the sport, I didn’t fully realize the extent of his love of the game.

I guess the love of tennis runs in the family, as in March I jumped at the chance to take a few days off from work and attend the sport’s “Fifth Major” at Indian Wells for the first time. After making the death-defying four-hour drive to the Coachella Valley, I knew immediately that the trip was worth it. The perfect weather, the windmills and mountains in the background, and tennis courts for as far as the eye can see—I have truly found paradise. Tennis paradise!

After familiarizing myself with the surroundings, I quickly made my way to the first-round match between Bernarda Pera and Daria Saville. With two evenly matched combatants, former U.S. Open champion Samantha Stosur in the coaching box, and doubles specialist Ellen Perez, freezing in flip-flops, sitting a row behind me, this was clearly the match to spectate.

And the match didn’t disappoint! But as we moved deeper into the third and final set, Mother Nature decided to take center stage. First came the clouds, then the sudden drop in temperature, and then the wind started howling. Lobs that would have gone in by 20 feet on a normal day started to fall short of the net—conditions forced normal tennis strategy out the window.

Shortly after the match concluded, Mother Nature told us that she wasn’t done: It started to rain. Not a drizzle, but a real storm. Fair to say, at this point no one knew what to do. Tournament organizers felt so confident about the typically pristine conditions that they didn’t even have covers for the courts, let alone a retractable roof.

As a veteran who has attended many tournaments in the past, I knew to quickly find one of the few covered spots on the grounds to keep from getting wet. A few other fans had the same idea, as we all huddled underneath the stands of one of the courts. When I arrived, one spectator started talking about how tennis needed to amend their archaic scoring system. A second patron started to defend how matches got scored, and how we needed keep the old traditions alive. Amazingly a third person next to me said, “Wait, aren’t you that puzzle guy? I bet you could make a great puzzle regarding tennis…”

**Challenge accepted!**

For all parts below, feel free to assume that we have two equally matched tennis competitors, and that there is no correlation between points scored. Also feel free to assume that a server wins 60% of the points on her serve.

1. The first player to win four points (by at least two points) wins a game. What is the probability that the person serving wins the game?

2. With players alternating serve each game, the first player to win six games and lead by at least two games wins the set. If players are tied at six games each, the set goes to a tiebreak. What is the probability that a set goes to a tiebreak?

3. In a tiebreak, after player A serves one point, each player then serves two times in a row for the remainder of the tiebreak. The first player to reach seven points and is ahead by at least two points wins the tiebreak. What is the mean number of points in a tiebreak?

4. Given that player A wins the set, what is the probability that player A won fewer points than player B in the set?

**Solutions may be emailed to puzzles@actuary.org. In order to make the solver list, your solutions must be received by August 1, 2024. **

**Solutions From Previous Issue: Arithmetic Formulas**

In this problem set, you were given at least 22 puzzles, of which you were to solve at least 11. In each of the puzzles, your input was three specified digits. Your goal was to find a formula, using each and only the input digits once and only once, to produce a specified output. For some puzzles, more than one distinct formula worked. You got credit for each, but you didn’t need to find them all.

In Section I, you were to use only the following arithmetic operations: sum (+), difference and negation (–), multiplication (*), division (/), exponentiation (^), decimal point (.), parentheses ((,)), and concatenation (adjoining two digits, e.g., taking 1 and 2 to make 21 or 1.2). Decimal points and concatenation could only be applied to digits; all the other operations could also be applied to expressions.

In Section II, three more operations could be used:

**Factorials:** Factorials could only be applied to
integers, that could include expressions equal to an integer; 0!=1; the
factorials of 1 and 2 add nothing so they couldn’t be used. The symbol !! was
interpreted to mean the factorial of a factorial, i.e., 3!!=6! (this differs
from standard practice in number theory).

**Roots:** Using the root symbol ( without an index meant square root but 2 was not to be taken from the input set; the index could be any expression other than 2. When using **∛**, a 3 was needed from the input set, etc.

**Repeating decimals:** In repeating decimals, a line is
drawn under digits to show that they repeat endlessly. For example, .7=.777… Repeating decimals could only be
applied to digits.

These puzzles were adapted from *Mental Gymnastics:
Recreational Mathematics Puzzles *©2011 by Dick Hess and first
published by Dover Publications, Inc.

**Solvers**

*Bob Conger, Deb Edwards, Bill Feldman, Yucheng Feng, Rui
Guo, Clive Keatinge, Jerry Miccolis, Don Onnen, David Promislow, Anna Quady,
Anthony Salis, Al Spooner, and Daniel Wade.*