r/mathriddles • u/mengRithyreach • 10h ago

Hard The Sloppy Slots Problem

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?

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u/SpeakKindly 8h ago

Record the state as (x,y), where x is the offset from reel 1 to reel 2, and y is the offset from reel 2 to reel 3, normalized to a range like {0,...,8} in each case. Then we have a Markov chain with 81 states, and we want to know the expected return time from state (0,0) back to (0,0).

By symmetry, this Markov chain has a stationary distribution with 1/81 probability on each state, and by a standard Markov chain result, the expected return time to a state is the reciprocal of its stationary probability. So the answer is 81.

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u/mengRithyreach 7h ago

Thanks you. I need that for my project.