Gear Loops

Level pending

Here we see on the back of a two pound coin nineteen gears all linked in a "gear loop". Let us say that there are n n gears in a certain "gear loop". What is the requirement for n n such that all the gears can turn (they can all mesh)?

Clarification: All gears turning means that if we turn one gear clockwise, the entire system will move.

No value of n n works. n n is a positive even integer. n n is a positive integer. It depends on the number of teeth on each gear. n n is a positive odd integer.

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1 solution

Arul Kolla
Apr 17, 2017

Let us label the gears as 1 , 2 , 3 , , n 1, 2, 3,\cdots, 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 n . If it is turning clockwise, then we will have a contradiction, because there will be two gears (namely n n and 1) that are touching but are going in the same direction. Thus, gear n n must be going opposite to gear 1. This only happens when n n is even.

Note: The back of the two pound coin will not turn!

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