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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

For the first weighing you need to put 4 weights on each tray.

Now look at each possible outcome of the first weighing and decide what you would do in each case.
You might like to first consider the case when the scales balance on the first weighing - what do you know about all the weights on the balance?

You might also like to think about the problem with fewer weights first, and use ideas you come up with there for the 12 weight problem.