Suppose you have 100 light bulbs wired in series with a button.
After the first button press all the lights turn one.
After the second button press every other bulb turns off.
After the third button press, the bulbs change state.
After 100 button presses, which bulbs are on?
After the first button press all the lights turn one.
After the second button press every other bulb turns off.
After the third button press, the bulbs change state.
After 100 button presses, which bulbs are on?
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If you're asking the question that people normally ask regarding the 100-bulb scenario, you missed a critical piece of information. (i.e., after the third press, every third bulb changes state, then after the fourth press, every fourth bulb changes state, etc.) If that's not what you're asking, you didn't provide enough information, because nothing in the problem as stated suggests what should happen for the fourth or subsequent presses.
If it's the problem that I think it is, here's the answer and explanation:
The bulb numbers that are perfect squares are on, and all the rest are off. That's because all the other numbers have only corresponding pairs of distinct factors, so the press corresponding to the first factor will turn the bulb on and the press corresponding to the second factor will turn the bulb off. But in addition to potentially having other pairs of factors, a perfect square also has its square root as a factor, and that only gets visited once. Hence the on-off pairs of button presses are supplemented by one additional single button press for the square root, which turns the light on.
Therefore, the bulbs that remain on are:
1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
If it's the problem that I think it is, here's the answer and explanation:
The bulb numbers that are perfect squares are on, and all the rest are off. That's because all the other numbers have only corresponding pairs of distinct factors, so the press corresponding to the first factor will turn the bulb on and the press corresponding to the second factor will turn the bulb off. But in addition to potentially having other pairs of factors, a perfect square also has its square root as a factor, and that only gets visited once. Hence the on-off pairs of button presses are supplemented by one additional single button press for the square root, which turns the light on.
Therefore, the bulbs that remain on are:
1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
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All of them. There are 100 presses, which means (33 sets of all three presses listed above) + 1 press. And since the first press of each set turns all the bulbs on...