By Josh Feldman
In February, I took my annual trip to Las Vegas and attended my first-ever “professional” pickleball match—though I am not sure professional is the right word, since most of the players earned their fame and fortune on the tennis court. But still, how could I pass up the opportunity to see Andre Agassi, Andy Roddick, John Isner, and Steffi Graf? Amongst my friends, people either seem to really love or really despise pickleball. Having never really seen or played the sport, I wanted to keep an open mind about pickleball before passing judgement.
And it turned out I had a blast at the Pickleball Slam! I even enjoyed the sound of the pickleball hitting the racket. In fact, the continuous clanks of the ball had this mesmerizing effect, almost putting me in a deep sleep. I felt like I was in a dream-like state, and I thought about all the puzzles I have written over the years. Has it really been 10 years that I have been doing this? Ten Years?!allall How the heck have I not run out of ideas by now? Sure, there have been a few times other people have given me some much-needed inspiration, but the majority are of my own. I also thought about how so, so many times I really do rely on my puzzle-reading friends to come up with the solutions to these outrageous problems. I truly don’t thank the puzzle solvers enough.
After an errant shot ended the point, I immediately got out of my trance, and suddenly realized I have another puzzle to compose. And then the emcee of the Pickleball Slam started explaining for what must have been the eighth time the scoring system of the event. I have no idea what it is about racket sports, but each one seems to outdo the other in coming up with the most convoluted scoring system imaginable. In this event, the first team to 15 points wins. It doesn’t matter if you serve or receive, whoever wins the point gets a point added to their score and will also serve the next point. But there’s a catch; there always seems to be a catch. You can only get the fifteenth and final point on your serve. Hence if you are winning 14 to 5 and you win the point on your opponent’s serve, the score remains 14 to 5 but now you get to serve for the match.
A random dude sitting next to me openly complained that the rules were rigged in order to produce a tight match, and sensing that I was a math person, asked me what the probability is of the match ending with the exact, desired score of 15 to 14. Just like in every puzzle column I write, I told him while I don’t know the answer off-hand, I bet my puzzle-solving friends do.
For each question below, assume that each pickleball point is random with no correlations and that both the server and receiver have an equal chance of winning every point.
- What is the probability that a match will end up with the final score being 15 to 14?
- What is the mean number of points played during the match?
- When I got home and did some research on pickleball, I discovered that receiving actually offers a slight advantage—the receiver wins about 55% of the points, not 50% as originally assumed. How does this new information change the answers to questions 1 and 2?
Solutions may be emailed to puzzles@actuary.org.
In order to make the solver list, your solutions must be received by August 1, 2025.
More Odds & Ends Solutions
These problems look easy and yet most solvers missed at least one of them on their initial submissions. It is important to read problems carefully.
Problem 1
There are 10 different positive integers; exactly 5 of them are divisible by 5 and exactly 7 of them are divisible by 7. Let M be the largest of these 10 integers. What is the smallest possible value of M?
- Answer: 70
- Solution: There must be at least two integers divisible by both 5 and 7. The smallest of these are 35 and 70. Both must be included; the remaining can be less than 70.
Problem 2
How many positive integers less than 1000 can be represented as:
a) The product of two odd numbers?
b) The product of an odd and an even number?
c) The product of two even numbers?
- Answers: a) 500, b) 499, c) 249
- Solutions: For a) and b) since 1 is odd, we want the number of odd integers and even integers less than 1000, respectively. For c) the number of integers divisible by 4.
Problem 3
How many two-digit numbers are there whose two neighbors are a prime number and a perfect square? List them.
- Answer: There are 5: they are 10, 24, 48, 80, 82.
- Solution: Prime must be >2, so it is odd. Square is prime ±2, so it also must be odd. So square is in {9, 25, 49, 81}. For each we test whether square ±2 is a 2-digit prime.
Problem 4
There is a 4 × 4 square that is partitioned into sixteen 1×1 square cells. From a given cell you may move only horizontally or vertically to another cell, but you cannot move to a cell next to your current cell nor to a cell you have been to before. What is the maximum number of cells (including your starting cell) that you can reach starting from
a) One of the four corner cells? Show your path.
b) One of the eight edge cells?
c) One of the four center cells?
- Answers: a) 16, b) 16, c) 16
- Solution: Experiment. A few solvers found a Hamilton cycle solving all parts at once.
Problem 5
There is a strange island with exactly 9 boobies: some red-footed and some blue-footed. When 3 of the boobies happen to meet, there is a 2 in 3 chance that none of them is a red-footed booby. How many of the boobies on the island are blue-footed? Explain.
- Answer: 8
- Solution: 8 is the only integer solution, n, to ()/() = .
Solvers: Bob Alps, Bob Conger, Andrew Dean, Deb Edwards, Bill Feldman, Rui Guo, Steve Itelson, Clive Keatinge, Ethan E. Kra, Sharon Kuester, Douglas A. Levy, Don Onnen, David Promislow, Anna Quady, Lenny Shteyman, Al Spooner, Matt Stephenson, Daniel Wade, Abraham Weishaus, Peter Whipple.
Apologies to Clive Keatinge for not including him as a solver to the previous puzzle.