The Backstory
I came up with this puzzle variant after watching a video posted on social media of two girls racing to put cups in a bucket. The winner was clearly due to a faster player but made me think what if I took that variable out and introduced something less obvious.
And so here it is, the velocity and the distances are all the same; instead I introduced a unique setup, tested it across trials, worked out the solution, mathematically demostrated the outcomes, then came up with an algorithm that scales up or down to any number of cups for any distances. I call It: Tony’s Cup Drop Paradox!
Introduction to the Puzzle
Imagine two players standing at opposite ends of a straight path, each tasked with collecting and dropping 10 cups into their designated buckets. The setup seems simple enough but the outcome may surprise you!
Here’s the setup:
Two players compete (let’s call them Player A, Player B). There are 10 cups for each player arranged in their own lanes. There are two buckets at the far left side of the “court” (let’s call the play area a court), one (Bucket A) for Player A, and another (Bucket B) for Player B. This is where each player must drop their cups into after picking up their cups.
Player A starts at the finish line, where both buckets are located. She moves outward, picking up each cup one by one and returning it to Bucket A, repeating this process until all 10 cups are collected.
Player B, on the other hand, begins at the far end—away from the finish line. She first grabs the cup closest to her (which happens to be the farthest from Bucket B) and drops it in. Then she moves to the next nearest cup (nearest to her original starting line) and continues this cycle until all cups are placed in Bucket B.
Each cup is spaced exactly 1 foot apart, with the first cup for each player positioned 1 foot ahead of their respective starting lines. Both players move with equal speed and accuracy, and fatigue is not a factor.
Who will finish first? And can you prove it?
The challenge comes down to pure mathematical reasoning! Welcome to the Cup Drop Paradox where intuition and logic collide!
Visual representation of the setup:
The cups in orange and purple are laid out in rows with each adjacent cup at equal distance from another: 1 foot apart. The orange cups are for Player A to pick up (one by one) and drop into the orange Bucket A (one by one). The purple cups are for Player B to pick up (one by one) and drop into the purple Bucket B (one by one). As you can see from this court layout, Player A starts from the left and moves to the right to pick up the first cup (Cup #1), then return to (moving left) Bucket A to drop it in, then moves on to pick up the second cup (Cup #2), then return to (moving left) Bucket A to drop it in, then moves on to pick up the third cup (Cup #3), until the 10th cup is picked up and dropped into Bucket A.
Similarly, Player B has to accomplish the same but starting from the right side of the court, picking up her first cup (Cup #10 since we’re counting along the x-axis, left to right) which is closest to her and travel left toward her Bucket B to drop it off, then return (moving right) to pick up the next cup (Cup #9) and doing the same until all of her 10 cups are dropped into Bucket B.
Think you’ve figured it out? Before you say: “it’s obvious,” take a closer look—this paradox might just surprise you. Find the solution, drop your answer in this form: https://forms.gle/TRS6nexfRZHpaEoo9 (anonymous survey) and back it up with math 🙂
After your submit your responses via the link above, you can also see how many other responses were submitted and their answers (anonymous). I will post the real answer along with detailed proof soon after giving you a chance to have fun with it! See you back here soon!
