I’ve seen this video where GOOGOL gear is depicted (it takes 10100 years on reaching the final gear) but how many generations will it take? The 5th gear takes about 10 hours in real time, by the time the 10th one starts spinning (you’re probably already dead).
The entire thing consists of 100 gears. However, even if you record or livestream a video of the entire thing: would you still be alive by the time the final gear starts spinning? The common life span of a human is around 72-73 median but some can reach 100+ up to 125.
You would need to spin the first gear 1 trillion times per second from now until the last proton in the universe decays, then do that about 1068 more times. 1068 is roughly equal to 52!, the number of unique ways a deck of cards can be shuffled. There’s a somewhat famous description of how large this number is (paraphrased for brevity):
Start a timer that will count down the number of seconds from 52! to 0. Every billion years, take a step along the equator. Every time you go around the world, remove one drop of water from the Pacific Ocean. Every time you empty the ocean, take one sheet of paper and place it flat on the ground. Every time the stack of paper reaches from the Earth to the Sun, take a step up Mount Everest. Once you’ve reached the summit 80 times, the timer will have reached 0.
I think most people don’t understand how insanely, mind-mindbogglingly large 52! is.
I’ve received disbelieving replies before after saying that every freshly full-shuffled deck, where that shuffling started from a shuffled state (not new deck order), is in a unique card order never before used by humanity since the invention of the four suit and 52 card deck.
But that statement is absolutely true. The chances of it being false are so microscopically tiny that they are, for all intents and purposes, zero.
Starts spinning? If they’re all connected, technically the moment the quickest gear turns, they all turn at the same time, just each more slowly than the next.
That said, the outer edge of the last gear is being moved by the mechanism at less than a Planck length per second, so it’s actually being moved more by the environment than it is by anything else, so in a way you could say it isn’t moving at all. But it still differs from a gear that is not in the mechanism at simply because it is intrinsically linked to everything else in a very specific way.
Now, if by “starts spinning” you mean “appears to have moved”, that’s a subjective assessment. Eyesight will play an important role. Some people might be able to discern a thousandth of a degree of difference between a fixed mark and a mark on the outer edge of a gear. Are we allowed to use external magnification? What about a scanning electron microscope?
There’s a version of this where the last gear is embedded into a cement block that is bolted to the base of the machine.
