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There are $n$ cups labeled $1,\dots,n$, each with a water tap that adds water into it at the same rate. There are also $k$ buckets, and $k$ sets $S_1,\dots,S_k\subseteq\{1,\dots,n\}$. At any point, if there is one unit of water in the cups in $S_i$ combined, you pour all that water into bucket $i$. (If the condition is satisfied for two or more $i$ simultaneously, choose the lowest $i$ first.) As time goes on, what does the ratio between water in the $k$ buckets converge to (and does this limit even always exist)?

As a concrete example, suppose $n = 7$, $k = 3$, $S_1 = \{1,2,3,4\}$, $S_2=\{4,5,6\}$, $S_3=\{6,7\}$, then the ratio between the water in the three buckets converges to $9:4:4$. This can be observed via a computer program and then proved formally, with heavy calculations again based on a computer program. Is there a way to determine an answer based on $S_1,\dots,S_k$ without relying on a computer?

There are $n$ cups labeled $1,\dots,n$, each with a water tap that adds water into it at the same rate. There are also $k$ buckets, and $k$ sets $S_1,\dots,S_k\subseteq\{1,\dots,n\}$. At any point, if there is one unit of water in the cups in $S_i$ combined, you pour all that water into bucket $i$. (If the condition is satisfied for two or more $i$ simultaneously, choose the lowest $i$ first.) As time goes on, what does the ratio between water in the $k$ buckets converge to (and does this limit even always exist)?

As a concrete example, suppose $n = 7$, $k = 3$, $S_1 = \{1,2,3,4\}$, $S_2=\{4,5,6\}$, $S_3=\{6,7\}$, then the ratio between the water in the three buckets converges to $9:4:4$. This can be observed via a computer program and then proved formally, with calculations again based on a computer program. Is there a way to determine an answer based on $S_1,\dots,S_k$ without relying on a computer? Can we set up some kind of "steady-state" equations?

There are $n$ cups labeled $1,\dots,n$, each with a water tap that adds water into it at the same rate. There are also $k$ buckets, and $k$ sets $S_1,\dots,S_k\subseteq\{1,\dots,n\}$. At any point, if there is one unit of water in the cups in $S_i$ combined, you pour all that water into bucket $i$. (If the condition is satisfied for two or more $i$ simultaneously, choose the lowest $i$ first.) As time goes on, what does the ratio between water in the $k$ buckets converge to?

As a concrete example, suppose $n = 7$, $k = 3$, $S_1 = \{1,2,3,4\}$, $S_2=\{4,5,6\}$, $S_3=\{6,7\}$, then the ratio between the water in the three buckets converges to $9:4:4$. This can be observed via a computer program and then proved formally, with heavy calculations again based on a computer program. Is there a way to determine an answer based on $S_1,\dots,S_k$ without relying on a computer?

There are $n$ cups labeled $1,\dots,n$, each with a water tap that adds water into it at the same rate. There are also $k$ buckets, and $k$ sets $S_1,\dots,S_k\subseteq\{1,\dots,n\}$. At any point, if there is one unit of water in the cups in $S_i$ combined, you pour all that water into bucket $i$. (If the condition is satisfied for two or more $i$ simultaneously, choose the lowest $i$ first.) As time goes on, what does the ratio between water in the $k$ buckets converge to (and does this limit even always exist)?

As a concrete example, suppose $n = 7$, $k = 3$, $S_1 = \{1,2,3,4\}$, $S_2=\{4,5,6\}$, $S_3=\{6,7\}$, then the ratio between the water in the three buckets converges to $9:4:4$. This can be observed via a computer program and then proved formally, with heavy calculations again based on a computer program. Is there a way to determine an answer based on $S_1,\dots,S_k$ without relying on a computer?