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While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).

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Essentially, one has a distribution of coins on a table, and you get to drop one coin at a time at one end, which ends up being pushed into the table, thereby potentially pushing coins off the edge. Note that you can choose where you can drop your coin, width wise. For simplicity, assume coins cannot stack on each other.

My question is, are there known limit laws for this game? That is, if I specify a distribution of coins on the table, and then start dropping coins in randomly, what can be said about how the expected number of dropped coins fluctuates, per turn. Consequently, are there various phase transitions as a function of coin density? As well, if I feed coins at a specific spot, what will the distribution of coin falls look like as a function of the table width? Do the boundary conditions (the side walls and the pusher) create interesting "modes" in the coin falling distribution?

I would think that this has to do with sand stacking cascades and KPZ growth but, do not have much experience in this area. Or perhaps this is just a simple Galton box that produces a normal distribution?

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I'm not a math guy so frankly I have almost zero understanding of what you are asking but I can tell you as a person who operates these games they pay out 30-50% depending on how they are adjustment. the main mode of adjustment is the side holes where money falls into and goes into the bottom of the machine (this is the oeprators profit) can be opened and closed to adjust the payout, if the hole is mostly closed they will pay out close to 50% of what they take in, if itss open they pay out around 30% there isnt much trick to how you put in the quaters are far as i know. also the angle of the lip will adjust the pay out as well.

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