Why Only Square-Number Bulbs Stay On: A Simple Ruby Demo

My son, a third‑grade student, asked for help with an Olympiad‑style problem about 200 light bulbs numbered 1…200, each controlled by its own switch. All bulbs start off. First we toggle every bulb whose number is divisible by 1, then those divisible by 2, and so on up to 200. How many bulbs remain on?

Observing a few bulbs shows that a bulb ends up on if its switch is pressed an odd number of times. A bulb numbered N is toggled once for each divisor of N, so the question becomes: for which N does the divisor count become odd?

If N is not a perfect square, its divisors can be paired (d, N/d) with d < √N, giving an even count. If N is a perfect square, the square root pairs with itself, adding one unpaired divisor, resulting in an odd count. Therefore exactly the bulbs whose numbers are perfect squares stay on.

There are 14 perfect squares ≤ 200 (1², 2², … 14²), so 14 bulbs remain lit.

To illustrate the reasoning to my son I wrote a short Ruby script that simulates the toggling process. Running the script prints the numbers of the bulbs that stay on, confirming the mathematical result.