Examples of specific information gathering and pattern finding

question (e.g. on a test)

There are 100 lockers and 100 people. The lockers are numbered 1, 2, 3, ... , 100 and numbered.

The first person opens all the lockers.

The second person closes the locker every two, four, six, ... and close the lockers every two.

The third person closes the door if it is open, opens if it is closed, and closes it if it is closed. and every third one, closing if open, opening if closed.

After the fourth person, sequentially "Nth person opens and closes every N".

How many lockers are open after the 100th person opens and closes the 100th locker?

Write down the opening and closing of each of the 100 lockers and count the open lockers.

Let's say you have (appropriately) 10 lockers.

https://gyazo.com/b462a93aa00a5a4cdb9e22d6f2bf3047

After opening and closing them according to the procedure, the final three that are open are lockers 1, 4, and 9.

I don't know if it is generally true that "the lockers open after this work is done are square numbers."

If we want to try to prove this, we have to consider "why is the square number locker open?".

If you simply need just the number of the answer, there are 10 square numbers under 100, so you can answer that and be done with it.

As for "why are square numbers lockers open?" the person who has finished thinking about it can sum up the idea in a few words: "Only square numbers have an odd number of divisors."

But memorizing this sentence will not help you solve another problem.

If you don't think for yourself, you will never learn.

The key to solving this problem is not the knowledge that "only square numbers have an odd number of divisors," but the "way of thinking" that can get you there by looking at the problem statement.

---