Discussion: This student solves the problem by the use of parity and the invariance principle - investigating what changes and what stays the same as the allowed transformation is performed repeatedly. In this case, the sum of the weights is always changing by two and therefore if it began odd it must remain odd. This is inconsistent with all vertices ultimately being equal, since there are eight vertices. A second proof is given: the student looks at a set of four vertices that do not lie on any edges, for example, 1,2,3,4 in the given labeling and 5,6,7,8. If one adds the weights for these sets and subtracts then the initial value is 1. But for every transformation it remains 1 since 1 is added to a vertex of the first set and 1 to a vertex of the second set. With this method more can be proved, e.g. it is impossible to that at some point all the vertices have weights divisible by 3.

Written response: Your first method gives a direct proof but I like the second one better since it can be applied to many more questions, e.g. is it possible that at some time all the weights are divisible by 3 or 4 or even 37? All of these problems "fall" from this very beautiful application of the invariance principle.

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