Meathead Math - Rack the Wait

Puzzle for the week of 2/9/25

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 2/16/25). 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!

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I thought of this one while I was waiting for a squat rack.

A gym owner wants to determine the maximum number of patrons, K, who can work out simultaneously without significant waiting times for equipment. The gym has N unique machines, each designated for a specific exercise and usable by only one person at a time. Each patron plans to perform exactly 3 exercises, randomly selected from the N available options. For simplicity, assume all K patrons start and transition between exercises simultaneously. What is the probability that all K patrons can complete their workouts without waiting for equipment?

Before launching into the problem, note that there are two ways to interpret the scenario:

1. Local (Per-Round) Organization: Each patron moves to a piece of equipment on their list if it's available at each transition. If it isn’t available, the select a different piece of equipment on their list, and so on.

2. Global (Overall) Organization: A coordinator, with access to all patrons' exercise plans, schedules the exercise order to minimize conflicts. This approach can prevent situations where two patrons need the same equipment simultaneously in a later round, even though it was free earlier, indicating that a different ordering could have avoided the wait.

For this week will concern ourselves with the global case.

Please submit your thoughts, progress, or answers here.