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This was in a math puzzle book I own. The explanation made no sense to me. I'm curious if anyone can figure this one out. I'll post the answer to it sometime tomorrow or the day after. Good luck.

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SOD = sum of digits. Therefore, SOD(356) = 3 + 5 + 6 = 14.

What is SOD(SOD(SOD(4444^4444)))?

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The SOD operation is invariant mod 9.

Now, 4444 is equivalent to 7 mod 9.
And the totient of 9 is 6.
And 4444 is equivalent to 4 mod 6.
So 4444^4444 is equivalent to 7^4 mod 9.
Now, 7^4 is equivalent to 7 mod 9.

So SOD(SOD(SOD(4444^4444))) is equivalent to 7 mod 9.

Now, 4444^4444 has at most 4444*5=22220 digits which can be at most 9 each, so we're looking at SOD(4444^4444) must be less than or equal to 199,990. That means that SOD(SOD(4444^4444)) must be less than or equal to 45 (the largest SOD up to 199,990). Then SOD(SOD(SOD(4444^4444))) is going to be less than or equal to 12.

Now, we're looking for a number that is less than or equal to twelve, and equivalent to 7 mod 9. 7 is the only such number.