Add 10 balls, remove 1. Repeat infinitely. How many balls remain? The answer defies all intuition.
Imagine a vase and infinitely many numbered balls. We perform a supertask—an infinite sequence of operations completed in finite time:
• At 11:59:30 (30 sec before noon): Add balls 1-10, remove ball 1
• At 11:59:45 (15 sec before noon): Add balls 11-20, remove ball 2
• At 11:59:52.5: Add balls 21-30, remove ball 3
• Continue halving the remaining time...
At exactly noon, infinitely many steps are complete. How many balls are in the vase?
At step n, ball n is removed. So ball 1 is removed at step 1, ball 2 at step 2, ball 1000 at step 1000...
For ANY ball k: It was added at step ⌈k/10⌉ and removed at step k. By noon, EVERY ball has been removed!
Result: 0 balls
At each step, we remove the lowest-numbered ball present. Balls 10, 20, 30, 40... are never the lowest when we remove.
The key: Ball 10 is added at step 1, but we remove ball 1. At step 2, we remove ball 2 (lowest). Ball 10 survives forever!
Result: ∞ balls
Both strategies perform the exact same number of operations: add 10 balls, remove 1 ball, repeated infinitely. Each step nets +9 balls. Yet one strategy leaves zero balls and the other leaves infinitely many!
This shows that infinite operations don't behave like finite ones. The "net gain" reasoning (∞ × 9 = ∞) fails because we must track which specific balls remain, not just the count.
Infinitely many steps complete in exactly 1 minute!
Each bar represents a step's duration. Notice how they shrink exponentially!
Step 1: 30s | Step 2: 15s | Step 3: 7.5s | Step 10: 0.059s | Step 20: 0.000057s | ...
In Strategy B, every 10th ball survives forever. Watch them accumulate...
Current survivors in Strategy B: 0 (10, 20, 30, 40, ...)
The paradox was first formulated by mathematician John E. Littlewood in 1953, in his book A Mathematician's Miscellany.
It was later expanded by Sheldon Ross in 1988, who analyzed various removal strategies and their surprising different outcomes.
The concept of a supertask—completing infinitely many operations in finite time—was named by philosopher James F. Thomson in his famous "Thomson's Lamp" paradox (1954).
Some philosophers argue supertasks are physically impossible and thus the paradox has no resolution. Others see it as revealing deep truths about mathematical infinity.