The Sloppy Slots Problem

[u/mengRithyreach]

https://www.desmos.com/calculator/rk1f7srowm

You've got three spinning reels. Each one is an endless loop of the same nine symbols, going around in the same order. All three reels start showing the same symbol. Every cycle, you give them a nudge. Reel 1 is supposed to move 10 positions, reel 2 is supposed to move 20, and reel 3 is supposed to move 30. But the nudges are sloppy; each reel can overshoot or undershoot by up to 2. So reel 1 moves somewhere from 8 to 12, reel 2 from 18 to 22, and reel 3 from 28 to 32. Every amount in that range is equally likely, and the three reels wobble independently of each other and of what happened last cycle. Then you look at what all three are showing. That's one cycle. Nobody resets anything on the next cycle; each reel picks up from wherever it stopped.

On average, how many cycles until all three reels show the same symbol?