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# Ask the Slot Expert: Cascading reels slot math

1 June 2022

Question: I was doing research on par sheet related to cascading reels slots. I wonder how the cascading math is designed? Could you give me some idea?

Answer: I haven't been able to get my hands on a PAR sheet for a cascading reels slot machine, but this is my understanding of how they did the math. Once I tell you the secret, you'll say, "Of course."

The secret is that the reel layouts are fixed, just like on a slot that does not have cascading reels. When a winning combination causes the reels to cascade, the symbols on the virtual reel above the displayed symbols fall into the empty spaces.

Let's say we have a winning combination and we have two empty positions to fill on reel 1. First, any symbols already displayed on reel 1 fall down to fill empty positions below them. Now we have to fill the two empty positions on the top of reel 1.

One way to do that is to choose the symbols from a weighted pool of symbols. Some symbols are more likely to be chosen than others. Instead of knowing the symbol in the position, we have probabilities that a symbol is in that position. We could then go through all the probabilities of each replacement position's having each symbol. That's many computations.

Another method is that the replacement symbols are determined by getting another number from the RNG and using the symbols at that virtual stop. This procedure isn't any better than choosing from a weighted pool and is probably worse.

Let's try this method. We get numbers from the RNG that tell us where to stop each reel. We also remember the stop that is displayed at the top of each reel. Let's call it the High Water Mark (HWM).

When we need to fill two positions on reel 1, the symbols in the two positions above the current HWM for reel 1 fall into place and we raise reel 1's HWM by two.

You can imagine it this way. When the reels stop, all of the symbols on the reels above the displayed symbols are stacked above the reels, ready to fall into place if needed.

With this method, there's no random element after the initial stopping places are chosen. We know exactly which symbols will fall into the empty positions for every cascade.

Now all we have to do is iterate through each combination of stopping positions for the initial starting screen. Granted, there could be quite a few of them (e.g., 10,000,000,000 if each reel has 100 stops), but we have plenty of time.

We find any winning combinations displayed, note how much they're worth, remove the symbols in the winning combinations, drop the symbols already displayed and fill the empty positions with the symbols stacked above the reels. Now we go back to see if we have winning combinations again.

When we hit a dead end -- a cascade without a winning combination, that is -- we move on to the next combination of stopping points.

Keep in mind that a large percentage of the starting screens do not have a winning combination. We're done with them after evaluating the the starting screen. Maybe this won't take as long as we thought.

How do we handle free spins? We'll have to figure out the Expected Value of a spin. The free spins usually use special bonus reel layouts. It might be simpler to work with these reels.

If anyone can confirm the procedure I described above or correct it, please write.

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John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots