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There is a 'Maximum flow, minimum cut' theorem for which the proof is sketched in this question. Finding the solutions requires all cases to be checked.
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
Explore some of the different types of network, and prove a result about network trees.
Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.