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Magic Caterpillars

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

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Plum Tree

Label this plum tree graph to make it totally magic!

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Olympic Magic

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

The Great Weights Puzzle

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
This problem puts emphasis on trying to gain as much information as possible from each weighing, whatever the outcome. So if a weighing has one outcome which gives you a lot of information, but another outcome which doesn't gives you much new information, it is probably not going to be a useful way of identifying the odd weight.