Happy faces

My parents still have a basket of baby toys from the early eighties, and I came across this one recently.  These happy faces are linked in such a way that they can rotate together and don’t come apart.  Here’s a question to ponder: Beginning with the orientation in the photo, if you hold one of the gears still and rotate the other one halfway around it, will the smiley be upside-down, right side up, or neither?


  1. Nice puzzle, because it shows how our everyday mental model of gears is limited to wheels turning on a fixed platform, and how difficult it is to step outside that model. An easier way to think about it is: what if you held the connecting rod still and rotated one of the gears through a half-turn? Clearly both faces would be half-turned, so would end up with the same orientation. Similarly, if you keep one gear still and rotate the connecting rod half way round it (which amounts to the same thing), the moving gear will end up with the same orientation as the still gear: which is the right way up.


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