Monday, April 12, 2021

A Math Puzzle as a Network

There is a standard type of math puzzle that has been around at least since I was a child. The details vary, but the concept is consistent. You are typically given a few initially empty containers of various (integer) capacities, an essentially infinite reservoir of something that goes in the containers, and a goal (integer) for how much of that something you want to end up with. You have to figure out how to reach the goal without having any measuring instruments, meaning that your operations are limited to emptying a container into the reservoir, filling a container from the reservoir, or moving content from one container to another until you empty the source or fill the destination, whichever happens first. (All this is done under the assumption of no spillage, meaning the originator of the puzzle did not have me in mind.) I think I've seen a variant that involves cutting things, where your ability to measure where to cut is limited to stacking pieces you already have as a guide to the piece you want to cut.

A question popped up on Mathematics Stack Exchange about how to solve one of these puzzles using dynamic programming (DP) with backward recursion. The problem at hand involves two jugs, of capacities seven and three liters respectively, and a lake, with the desired end state being possession of exactly five liters of water. The obvious (at least to me) state space for DP would be the volume of water in each jug, resulting in 32 possible states ($\lbrace 0,\dots,7\rbrace \times \lbrace 0,\dots,3 \rbrace$). Assuming the objective function is to reach the state $(5,0)$ with a minimal number of operations, the problem can be cast just as easily as a shortest path problem on a digraph, in which each node is a possible state of the system, each arc has weight 1, and arcs fall into one of the categories mentioned in the previous paragraph.

I was looking for an excuse to try out the igraph package for R, and this was it. In my R notebook, a node label "5|2" would indicate the state where the larger jug contains five liters and the smaller jug contains two. Arcs are labeled with one of the following: "EL" (empty the larger jug); "FL" (fill the larger jug); "ES" (empty the smaller jug); "FS" (fill the smaller jug); "PLS" (pour the larger jug into the smaller jug); or "PSL" (pour the smaller jug into the larger jug).

Assuming I did not screw up the digraph setup, a total of nine operations are required to get the job done. If you are interested, you can see my code (and the solution) in this R notebook.

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