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It is well known that cats can be turned into perpetual motion machines under the right circumstances. Candy-sharing cats are such wonderful creatures that come in infinite supplies, labeled 1,2,3,...

An $M$ machine is made by the set of cats $\{1,2,3,...,M\}$ sitting in a circle, in any order, along with any distribution of candies among them.

  1. For any $n$, if cat $n$ has $n$ candies, it will share them by giving each cat in its clockwise direction one candy, until it runs out of candies or every other cat has got one from it.
  2. When one cat has finished the sharing, if there's another cat who qualifies for condition 1, it will repeat that process.
  3. If more than one cat qualifies for condition 1, the one with the smallest number will share.

If we can make an $𝑀$ machine in which there's always a next sharer, the candies will never stop flowing, then it's an $𝑀$ perpetual motion machine (M-PMM)!

An example of a 4-PMM is cat order $3,1,2,4$ and candy distribution $1,0,0,4$. It undergoes 6 sharings in a loop:

$3124$

$1004\\2111\\2021\\3002\\0113\\0023\\1004$

Question: Is there an upper limit to the size of a PMM?

Source & progress: I found the prototype of the puzzle in an obscure corner of the web, made some natural generalizations, and shared it on puzzling.stackexchange. We know 2,4,6,8-PMM exist, and 3,5,7-PMM don't. 10-PMM probably doesn't exist either, according to this.

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    $\begingroup$ Fantastic riddle, thanks Eric for sharing it!! $\endgroup$
    – Dominic van der Zypen
    Commented May 17, 2023 at 16:35
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    $\begingroup$ @DominicvanderZypen You're welcome, Dominic! The current known results are obtained more or less by brute force search, which becomes infeasible for more than 10 cats. $\endgroup$
    – Eric
    Commented May 19, 2023 at 6:16
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    $\begingroup$ Is the condition in step 1 that cat $n$ has exactly or at least $n$? $\endgroup$
    – Peter Taylor
    Commented May 23, 2023 at 15:07
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    $\begingroup$ That was a dichotomy. If it's "exactly" then that creates a lot of blocking states and may simplify the analysis; if it's "at least" then I would be inclined to consider whether a starting state assigning $n^3$ to each of them would work. $\endgroup$
    – Peter Taylor
    Commented May 23, 2023 at 16:19
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    $\begingroup$ There are definitely 10-PPMs. In fact, here's an exhaustive list of all PPMs with M<11. $\endgroup$
    – AnttiP
    Commented May 27, 2023 at 12:02

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