Meathead Math - Can you complete your drop set?
Puzzle for the week of 3/2/25
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 3/2/25). 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!
Your gym saga continues when your rival arrives and sets up on the bench press next to you. You start with a modest 5 lbs for your first set, but your rival—competitive and petty—insists on outlifting you and loads up 8 lbs. Not to be outdone, you decide to match his ego, adding his entire weight to yours for your second set, bringing your total to 13 lbs. Your rival follows suit, adding your previous weight to his, lifting 21 lbs.
At this point, your coach intervenes. Frustrated, he demands that you both start a drop set at exactly 1/2 of your current weight. However, you quickly realize the problem—weights can only be adjusted in whole-number increments, and your rival's odd weight (21) makes dividing by 2 impossible.
Your coach, determined to enforce the drop set, suggests exactly 1/3 of your weight instead. That doesn't work either. He tries 1/4, but once again, it's not possible. This continues through 1/7, and at this point, everyone recognizes that no larger fractions will work either. Frustrated, your coach gives up for now, telling you he’ll check back later. Until then, you and your rival can continue your escalating weight battle.
Assuming you and your rival keep adding weight in the same manner, what is the fewest number of additional sets needed before your coach returns and you both can perform a valid drop set?
What if you and your rival had decided to start with 5 lbs and 7 lbs instead?
Please submit your answers here. Please ask any questions in the comments.
Clarifying question: so we're trying to find two weights such that we can divide them both by some whole number, and both divisions will result in whole numbers, yes?