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This is a little puzzle I heard a while back from one of my mathematically inclined friends- I get the sense that it's bounced around a little, so forgive me if you've heard it.

There is a sort of function that, given any positive whole number, will return another positive whole number. Interestingly, there is one number that will return itself- the number 4. There are no other loops of any size, so you can start at any other number and, by repeatedly plugging in the output of the function, you will eventually reach 4.

Given this diagram showing a few examples of those succession chains, determine the function.

enter image description here

(If anyone feels like they really need more information, you can post a number in the comments and I'll give you its chain. However, the givens are sufficient to find a solution.)

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  • $\begingroup$ Can you upload that image to SE's imgur? I can't see it from where I'm at. You can do it through the 'edit' screen with that little button that looks like a fancy painting. $\endgroup$ – Bailey M Aug 6 '15 at 19:53
  • $\begingroup$ Should be fixed! $\endgroup$ – Patrick N Aug 6 '15 at 19:55
  • $\begingroup$ Not that I'll get a chance to solve it, but thanks. :P $\endgroup$ – Bailey M Aug 6 '15 at 19:56
  • $\begingroup$ You've had your fun today, Bailey. My turn! ;) $\endgroup$ – Roland Aug 6 '15 at 19:57
  • $\begingroup$ Well, you can always do so for your own enjoyment, but I don't think they award rep for having fun :P $\endgroup$ – Patrick N Aug 6 '15 at 19:58
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Four is cosmic because:

It has four letters.
Each number is followed by the number of letters in its spelling, but four leads back to itself indefinitely!
Four is the only cosmic number because it is the only number with its own number of letters in its English spelling, and there are no cycles of two or more words that have each other's letter counts.

And it should probably go: 18 --> 8 --> 5 --> 4 (from Doge's comment)

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  • $\begingroup$ Yup! I was a little afraid this would go fast, but I guess it is somewhat well-known (that, or you're a very fast solver!) $\endgroup$ – Patrick N Aug 6 '15 at 19:57
  • $\begingroup$ I've never heard it called "cosmic", but I definitely recognized the pattern from many years ago, don't know where exactly. $\endgroup$ – Roland Aug 6 '15 at 19:59

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