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I assume your students have seen Linear Algebra.
Remember in Linear Algebra how you sometimes have to solve $Ax = b$ for a matrix $A$ with more columns than rows? You usually get free variables, right? So after reducing to echelon form you have something like:
$$\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \...
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It occurred to me in thinking about the ménage problem that one could reformulate it as a story about students and advisors attending an awards dinner, with the requirement that advisors and students sit alternately, with no student sitting next to their own advisor. One must stipulate that no student has multiple advisors attending the dinner and that no ...
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Agents and missions are a good choice for maximum pairing problems for bipartite graphs. Edge linking agent to a mission means that agent can carry out the mission, so there are no edges linking mission to mission or agent to agent.
Agent can't do another agent, or we have an entirely different problem :)
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