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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. If we loosen condition 3 to allowing any qualified cat to share, instead of only the one with the smallest number, my guess is that won't change whether an M-PMM exists. But I would love to see a counter example.

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.

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 $3(1),1(0),2(0),4(4)$, where the number in the parentheses is the initial number of candies for the corresponding cat. It undergoes 6 sharings in a loop:

$3(1),1(0),2(0),4(4)\rightarrow3(2),1(1),2(1),4(1)\rightarrow3(2),1(0),2(2),4(1)\rightarrow3(3),1(0),2(0),4(2)\rightarrow3(0),1(1),2(1),4(3)\rightarrow3(0),1(0),2(2),4(3)\rightarrow3(1),1(0),2(0),4(4)\rightarrow ...$

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

I found the prototype of the puzzle in an obscure corner of the web, made some natural generalizations, and shared it on puzzling.stackexchange. Progress so far: we know 2,4,6,8-PMM exist, and 3,5,7-PMM don't.

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. If we loosen condition 3 to allowing any qualified cat to share, instead of only the one with the smallest number, my guess is that won't change whether an M-PMM exists. But I would love to see a counter example.

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 $3(1),1(0),2(0),4(4)$, where the number in the parentheses is the initial number of candies for the corresponding cat. It undergoes 6 sharings in a loop:

$3(1),1(0),2(0),4(4)\rightarrow3(2),1(1),2(1),4(1)\rightarrow3(2),1(0),2(2),4(1)\rightarrow3(3),1(0),2(0),4(2)\rightarrow3(0),1(1),2(1),4(3)\rightarrow3(0),1(0),2(2),4(3)\rightarrow3(1),1(0),2(0),4(4)\rightarrow ...$

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

I found the puzzle in an obscure corner of the web, made some natural generalizations, and shared it on puzzling.stackexchange. Progress so far: we know 2,4,6,8-PMM exist, and 3,5,7-PMM don't.

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 $3(1),1(0),2(0),4(4)$, where the number in the parentheses is the initial number of candies for the corresponding cat. It undergoes 6 sharings in a loop:

$3(1),1(0),2(0),4(4)\rightarrow3(2),1(1),2(1),4(1)\rightarrow3(2),1(0),2(2),4(1)\rightarrow3(3),1(0),2(0),4(2)\rightarrow3(0),1(1),2(1),4(3)\rightarrow3(0),1(0),2(2),4(3)\rightarrow3(1),1(0),2(0),4(4)\rightarrow ...$

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

I found the prototype of the puzzle in an obscure corner of the web, made some natural generalizations, and shared it on puzzling.stackexchange. Progress so far: we know 2,4,6,8-PMM exist, and 3,5,7-PMM don't.