Two children toggling a row of lamps

Question

Ten lamps stand in a row, numbered 1 to 10, and every one starts off. Two children walk past:

Alice flips (toggles) every lamp whose number is even.
Then Bob flips every lamp whose number is a multiple of 3.

Flipping means on→off or off→on.

When both children are done, which lamps are glowing?

Accepted Answer

A lamp starts off, so it ends on only if it gets flipped an odd number of times (once), and ends off if flipped an even number of times (zero or two). Each lamp can be flipped by Alice (if even), by Bob (if a multiple of 3), by both, or by neither. Go through 1–10 and count flips: - 1, neither → 0 flips → off - 2, even → 1 flip → on - 3, multiple of 3 → 1 flip → on - 4, even → 1 flip → on - 5, neither → 0 flips → off - 6, even and multiple of 3 → 2 flips → off - 7, neither → 0 flips → off - 8, even → 1 flip → on - 9, multiple of 3 → 1 flip → on - 10, even → 1 flip → on The glowing lamps are 2…

Two children toggling a row of lamps

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