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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).

alt text

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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What is the mathematical question here? – Stefan Kohl Jul 17 '15 at 16:53

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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I frequent a restaurant that has one of these machines. It has an upper level that moves toward the player then backwards. You can drop one quarter at a time through a movable slot so that you can drop several coins onto the upper shelf during one cycle.

The guys who work there have figured out pretty effective tactics and strategies for winning.

Strategy: Only play when quarters are built up heavily in the middle of the edge. (The sides don't matter.) The quarters should look like they will fall at any moment. Some may be largely over the edge but stopped from falling by other quarters lying on them.

Tactic: During one cycle, drop four quarters as close together as you can get them at the center of the machine. They act like a bulldozer to push coins in the center of the upper shelf forward. These, in turn, fall to the the lower shelf and push coins in the center toward the edge. This strategy minimizes movement of coins toward the edge slots and maximizes the flow of coins forward.

Sometimes they make quite a killing.

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