gears in a certain "gear loop". What is the requirement for such that all the gears can turn (they can all mesh)?
Here we see on the back of a two pound coin nineteen gears all linked in a "gear loop". Let us say that there areClarification: All gears turning means that if we turn one gear clockwise, the entire system will move.
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Let us label the gears as 1 , 2 , 3 , ⋯ , n . Let us turn gear 1 clockwise. Then gear 2 will turn opposite to that, gear 3 the same as gear 1, etc. Let us take gear n . If it is turning clockwise, then we will have a contradiction, because there will be two gears (namely n and 1) that are touching but are going in the same direction. Thus, gear n must be going opposite to gear 1. This only happens when n is even.
Note: The back of the two pound coin will not turn!