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So with a more expensive pizza, Charlie pays less even though the value of each slice increases?

This is a simple mechanics+ geometry puzzle.. I'm not sure if the "2015 MIT Mystery Hunt" is a type of metapuzzle?

It's not clear whether the back story to this post is the escape from constraint of anyone else's opinion, or simply the claim of absolute superiority of one's own thinking. This uncertainty is a meta-puzzle in itself, and it is one that may merit its own thread. However, this forum is for questions about logic and math, so if such a thread is started it should be placed in the Others forum. Further posts of this type are subject to deletion. Regarding the current thread, the author of the OP put "red ends" in his drawing, and he confirmed, when the question was raised, they are part of the required symmetry of the solving configuration.

No, I would have no reason to consider the [placement of the] heads, because it is a matchsticks puzzle. The matchsticks puzzles have never limited themselves to placement of the heads. If you had wanted that extra condition, then you are changing what kind of traditional puzzle this is and not announcing in the rules.