Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 6/15/25). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem! The solution and submitted responses will be posted around Wednesday at 10 am.
I make no guarantees my solutions are correct! You are all smart people so please comment if you think I made a mistake!
You and three of your friends are at the casino and have happened upon a new game. the game is still experimental so the casino is having a limited time offer where people can play for free and the winner receives $20. Seems like a no-brainer! You and your friends decide to play.
The game starts with an empty unit circle. Players place a point in the circle one at a time (player 1 first, player 2 second, etc.) until each player has gone. After everyone has selected, a random point is sampled uniformly from the unit circle. The player whose point is closest to the randomly sampled point is the winner.
Your friends decide to take the egalitarian approach and give each player an equal probability of winning. To accomplish this you all decide to select the center of one of the circle’s quadrants (think center of mass).
The first three players go and they each select a quadrant center. The last player—whom you’ve never fully trusted—decides to be greedy and select the point that maximizes his odds of winning. What point does he select and what are his odds of winning?
You and your two friends that just got duped are rather mad. The croupier managing the game offers for you all to play again on the condition that you retain the same order you played the first time. You all accept (it is after all free money at the end of the day) but the three of you decide you are going to conspire against the 4th player this time. The three of you need to figure out how to place your points such that player 4’s chances of winning are minimized.
Please submit your answers here. Please ask any questions in the comments.
Hey! As a heads up to readers, this puzzle series is going to culminate in a competition. I’m going to ask solvers to devise and implement a strategy of their own to play this game. Each person’s strategy will then be pitted against other submitted strategies to see who has the best! Strategies will likely take the form of a simple Python class. More details to come. The competition will benefit from lots of participation so I encourage everyone to start thinking about strategies for this game and—when the time comes—share the puzzle with anyone you think would be interested!
Really interesting problem! Game theory puzzles are always so fun to think about. This puzzle, while different, reminds me of the Jane Street Puzzle: "Robot Capture the Flag" https://www.janestreet.com/puzzles/robot-capture-the-flag-index/ .
What time zone is the due date?