T2 — 100 Doors Puzzle

Each door k is toggled once for every divisor of k. Only perfect squares have an odd number of divisors, so only those remain open.

Doors:
Open doors: —
Why perfect squares?
  • Door k is toggled on every pass d where d | k (d divides k).
  • Divisors come in pairs (d, k/d), giving an even count.
  • Perfect squares have one unpaired divisor (the square root), so they have an odd count → remain open.
Open = perfect squares Example (100): 1,4,9,16,25,36,49,64,81,100