Cups are placed on a table 3 facing up and 3 facing down If you flip 2 cups at a time is it possible to get all 6 cups facing up or all 6 cups facing down?

No. Every time you flip two cups one of three things must happen: you flip two that were up, so the number that are up decreases by two; or you flip two that were down, so the number that are up increases by two; or else you flip one of each, so the number that are up stays the same The number that are up is odd to begin with, and this stays the same after you flip each pair. You can have 1, 3 or 5 cups facing up, but never 6 or zero.

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