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I would appreciable any insight you give to solve this type of question:

How many $f: \{1,2,3\}\to \{1,2,3\}$ satisfy $f(f(x))=f(f(f(x)))$ for all $x$?

Normally questions of this type I used try by guessing, until I can't see any more possibilities. But I don't like this way of solving, do you know a theorem or some tool to solve this type of question with certainty?

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  • $\begingroup$ What have you tried? How many functions are there? If there are few functions, just list them all out and then see what conclusions you can draw. $\endgroup$
    – Calvin Lin
    Nov 22 '20 at 4:46
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(Fill in the gaps as needed. If you're stuck, explain what you've tried and why/where you are stuck.)

Hint: Show that we cannot have $f(a) = b, f(b) = a$.

Hence, we must either have A) $ f(a) = a$ or B) $f(a) = b, f(b) = c$.

Hence, there are $X$ such functions.

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