Meathead Math - Plate Permutation

Puzzle for the week of 2/16/25

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 2/25/25). 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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Now that we’ve got everyone sorted to their exercise with last weeks’ puzzle, you find yourself in the squat rack. Great, squat is one of your best exercises. In fact, in order to get a good workout, you need to add 5 plates to each side! Of course, you know from experience that you can’t just load each of the plates arbitrarily. In particular, if one side has ≥3 plates more than the other side, the bar will become too imbalanced and the weight will come crashing down (ignore the physics for now and just assume that this is true). How many unique ways are there to add plates to the bar so that each side has 5 plates and one side never has ≥3 plates more than the other side?

How about if we need to load up N plates? And what if the balance of bar in the rack is modified so that the bar will tip if one side as ≥m more plates than the other side?

Please submit your answers here. Please ask any questions in the comments.

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For those that attempted last week’s puzzle, I want to extend my condolences. It was tougher than I realized (my solution before posting was incorrect so I thought it was simpler than it was). I tried to dial it back a little for this week’s puzzle.

Also, if you’d like, you’re more than welcome to attempt to solve the variation of last week’s puzzle where there there is no global coordination for using the equipment. In this more realistic variant, people go to some piece of equipment on their list and if a patron finds that a piece of equipment they were going to use is occupied that round, they will select another of their exercises and try going to that corresponding piece of equipment and so on. Fair warning though, the problem (as best I can tell) ends up being conceptually similar to the global case but much more tedious. If there’s sufficient interest in a solution, I can write one up.