Math Archives - Slot Game Design

Going Wild: Slot Machine Math Tutorial 2

In Part 1 of this Slot Machine Math Tutorial series we showed how to create a PAR sheet to describe the math behind a simple slot machine game. The example game we used, the 1899 Fey Liberty Bell, had only one payline and no Wild symbols.

Now let’s explore how the math changes when we introduce Wild symbols to the game.

They’re Called Wild for a Reason

Wild symbols add excitement to a slot game because they increase the Hit Frequency and lead to more wins.

Wild symbols can substitute for any other symbol in line combinations — though many games make exceptions to this rule, saying Wilds cannot substitute for Scatter or Jackpot symbols.

As a result, there are usually a lot more winning combinations when Wilds are involved. The game’s RTP will go up as well, unless you adjust the payouts accordingly.

A Wild Liberty Bell Example

Wild Liberty Bell Slot Machine Math Example, Reels and Symbols — *Wild Liberty Bell Example, Reels and Symbols*

Let’s start with a very simple example: I created a Wild Liberty Bell PAR sheet (see the second tab) that adds one WILD symbol to the center reel of the old Liberty Bell. In this game, Wild can substitute for any other symbol, with no exceptions.

The Reel Strips and Symbol Distribution sections of the sheet are similar to before, with the addition of one WILD symbol at slot 9 on reel 2.

The Pay Table did not change in this version, but the formulas in the Line Combinations sections did.

Before we had the Wild symbol, the formula for the 3 Bells combinations was simple, where F22, G22, and H22 represented the number of Bell symbols on reel 1, 2, and 3 respectively:

F22 * G22 * H22 = 1 * 1 * 1 = 1

Now that there are Wilds on Reel 2, the effective number of Bell symbols on Reel 2 needs to include the Wilds:

F22 * (G22 + Wilds2) * H22 = 1 * 2 * 1 = 2

The same adjustment applies to all the combination formulas, increasing the Ways to Win and Total Payout across the board.

Wild Liberty Bell, Pay Table — *Wild Liberty Bell Example, Pay Table and Ways to Win Combinations*

The Hit Frequency has increased to 27.4% (compared to 26.4% for the original Liberty Bell) and the RTP has gone way up to 88%, compared to 75.4% for the original. That’s a whopping 12.6% increase in RTP just from adding a single Wild symbol to one reel.

Where to Put the Wilds?

When you’re designing a slot game that includes Wilds, you have some basic decisions that can greatly affect the complexity of your game. The decisions involve whether to:

- Include Wilds on the first reel;
- Include Wilds in Jackpot combinations;
- Make the Wild symbol the Jackpot symbol;
- Use Wild multipliers.
  All Wild Liberty Bell Example, Reels and Symbols

Wilds on the First Reel

The first time you’re designing a slot game, it’s tempting to just put Wild symbols on all the reels and see what happens. But adding Wilds to the first reel greatly increases the complexity of the slot math, the PAR sheet, and the software that will be coded to evaluate line combinations.

I created another example PAR sheet to illustrate how the formulas can change when you also put a Wild symbol on Reel 1. This “All Wild Liberty Bell” variant has one WILD symbol per reel. (The RTP is over 130%, because the pay table did not change, but this is example only for illustration purposes).

The main question to answer is: how should the game handle “All Wild” combinations, where the Wild symbol lands on the payline for every reel?

The rule of thumb for “all wild” combinations, in a game where the Wild symbol is not the Jackpot symbol, is that it should represent the highest-paying matching combination. In this example, Wild-Wild-Wild would be treated as Bell-Bell-Bell and pay 20.

The formula for the Bell-Bell-Bell outcome isn’t much more complex than before:

(F22 + Wilds1) * (G22 + Wilds2) * (H22 + Wilds3) = 2 * 2 * 2 = 8

But the formulas for all the other combinations become much more complex, because now they must subtract the higher-paying Wild-Wild-Wild combination. Here’s the formula for 3 Hearts:

((F22 + Wilds1) * (G22 + Wilds2) * (H22 + Wilds3)) - (Wilds1 * Wilds2 * Wilds3) = (2 * 2 * 2) - 1 = 7

Now, consider that this is a very simple three reel, one line slot. When designing a five reel slot with 2, 3, 4, and 5 symbol winning combinations, the formulas get extremely complex when you have Wilds on the first reel.

*All Wild Liberty Bell Example, Pay Table and Ways to Win Combinations*

My advice for first-time slot designers is to avoid putting Wilds on the first reel, at least until you are comfortable using simpler Wild-substitution formulas.

Wilds in Jackpot Combinations

The simplest rule for Wilds is to let them substitute for any symbol, including the Jackpot symbol. (Scatter symbols are normally an exception.)

This usually means you will have a lot more Jackpot winning combinations, and therefore the Jackpot payout will have to be smaller. If you add a lot of Wild symbols to the reels — perhaps to increase the Hit Frequency — this can lead to a rather small and unexciting Jackpot payout.

One solution is to exclude the Jackpot symbol from Wild substitutions. Then you keep the number of Jackpot combinations small and maintain a high jackpot payout.

Keep in mind that most players won’t read game rules or pay tables before playing your slot. If you exclude some symbols from Wild substitution rules, expect to get feedback from a few upset players who got a “near miss.”

Wild as the Jackpot Symbol

Many slot games make the Wild symbol the Jackpot symbol, so that a line full of Wilds will have the highest payout. The math on this variation can work out well, especially when you don’t have a lot of Wilds on each reel. You need to put Wilds on the first reel for this to work, so get ready to handle the extra complexity.

Wild Multipliers

Some games treat each wild symbol in a combination as a multiplier. For example, a 2X Wild symbol would double the line’s payout. The main consideration for this feature is whether multiple Wilds on a line are additive or multiplicative. For example, if a line has three separate 2x Wild symbols, is that a 6x multiplier or a 8x multiplier?

Needless to say, this feature leads to huge line payouts, and the math in your PAR sheet will be relatively complex. I’ll cover Wild multipliers in more detail in a future post.

Main Takeaways

Here are some important things to consider when adding Wild symbols to a slot game:

- Adding Wilds is a great way to increase Hit Frequency.
- Wilds also have a big impact on RTP, so be prepared for lower line payouts on average.
- Don’t put Wild symbols on the first reel, unless you’re ready for very complex combination formulas (or unless you want the Wild symbol to be the Jackpot).
- If you want to keep Jackpot payouts high, don’t let Wilds substitute for Jackpot symbols.

Creating PAR Sheets, Slot Math Tutorial 1

What is a PAR Sheet?

A PAR sheet is a spreadsheet-like tool for defining the math model of a slot machine game. PAR is an acronym for Probability and Accounting Report. PAR sheets were originally used when designing physical slot machines. Of course today they are just as effective when designing slot games for online and social casino apps.

Fey Liberty Bell Slot — The classic Fey Liberty Bell (circa 1899).

Historically, slot machine manufacturers have zealously guarded their PAR sheets as trade secrets. It isn’t easy to find examples of actual PAR sheets for classic slot machine games online. Some have leaked out, but often it has required researchers to petition the manufacturers using Freedom of Information Act laws.

I’ll try to demystify PAR sheets in this and future posts, and show how valuable they are for designing slot machine games.

There is no standard format for a PAR sheet, though they usually include the same things, depending on the game features:

- A summary of key performance factors like Return to Player (RTP), Hit Frequency, and Volatility
- Reel Symbols and Strips
- The Pay Table for line wins
- Wild symbol probabilities and payouts
- Scatter symbol probabilities and payouts
- Free Spin features
- Bonus Game features
- Other special features, such as multipliers or expanding wilds

Par sheets for physical slot machines might include additional data about physical limitations or features of the mechanical reels.

A Simple PAR Sheet Example

A PAR sheet presents the theoretical probabilities and payouts for a slot machine game. Generally it lists all the possible combinations of reel positions on a slot, determines how many of those represent “winners” vs. “losers,” and then calculates the overall RTP (also known as payback percentage) from that.

I’ll start with a simple example for a three-reel slot machine with one payline. This is a PAR sheet for one of the all-time classic slot machines, Charles Fey’s Liberty Bell. Released in 1899, the Liberty Bell had a number of technical innovations and became the most popular slot machine of its day. It’s a great example of a simple three-reel, one-line slot.

Summary and Performance Statistics

In the first section of the PAR sheet I like to summarize the basic rules and features of the game, and the key performance indicators,

This acts like an executive summary of the game design, and makes it easy for reviewers to see the key features and values at a glance.

Reel Strips and Symbol Distribution

The next section lists all of the slot symbols as they appear, in order, on each of the reels. They are indexed as “stops,” a term which refers to the possible stopping positions of the mechanical disks used on physical slot machines.

The order of the symbols on the reel can make a big difference. In multi-line slot machines, for example, ordering symbols differently on the reels can influence the number of line wins per play.

In the single payline Liberty Bell game, the order does not affect the math, but it does reflect the actual reel strips that Fey used in his classic machine.

The Symbol Distribution section counts the number of each type of symbol on each reel, to make the combinatorial math formulas simpler in the next section. It also shows the total number of symbols per reel, which are multiple to show the Total Ways.

The Liberty Bell has 10 symbols on each reel, meaning there are 1,000 Total Ways, or possible combinations of the three stop values.

Pay Table and Ways to Win

The Pay Table lists all of the winning symbol combinations and how much they pay, in terms of a multiplier of the bet unit.

Most PAR sheets assume a bet unit of one credit, to simplify the math involved. This is fine for simple games, as long as all of the payouts scale exactly with the bet amount.

In a more complicated example the Pay Table might have its own tab or section in the spreadsheet. In this example, the Pay Table and Line Combinations sections are combined.

The Pay Table lists each of the 3-symbol combinations and its associated payout value.

The real work happens in the Line Combinations section. The Ways to Win column multiplies the values from the columns in the Symbol Distribution section to determine how many combinations are theoretically possible for each winning outcome. The formulas for the Liberty Bell are quite simple, as you can see in the spreadsheet.

Now we can calculate two important values. First is the Hit Frequency, which represents how often a player will win something from a play of the game. This is a one payline game so the Hit Frequency formula is simply:

(Total Ways to Win)/(Total Ways) = 264/1000 = 26.4%

Another way to look at Hit Frequency is in terms of how many spins, on average, it takes for a player to win something. For the Liberty Bell this is roughly one win every four spins:

1/(Hit Frequency) = 3.79

In this game, all the line payouts are higher than one credit, but in other games the Hit Frequency also includes wins that are less than original bet amount. We’ll talk about the effects of those kinds of “false wins” in future posts.

The other key value is the Return to Player (RTP), which is the amount that a player can expect to win on any given spin.

(Total Payout)/(Total Ways) = 756/1000 = 75.6%

An RTP of 75.6% indicates that for every dollar a player put into a Liberty Bell machine, they could expect to win roughly 75 cents in return.

These formulas take an average of all the possible winning line combinations and line win payouts to arrive at these figures. Theoretically, it is like a player spun this machine 1,000 times (the Total Ways) and received each possible combination of stops one time, also known as a full cycle. (This concept can be applied later if you write programs to simulate slot game outcomes and validate the PAR sheets.)

The RTP is one of the most important values to present in the PAR sheet. In most cases when designing a new slot game you will target a specific RTP range (or ranges), while also balancing Hit Frequency, Volatility, and other factors. We’ll discuss all of these factors in more detail in future posts.

Underlying Assumptions

This is a very simple example of a PAR sheet but it illustrates some core assumptions:

- A PAR sheet usually assumes a simple bet of one credit per play.
- Ways to Win calculations are defined for only a single payline. This is true even for multiline games — though there are exceptions to this rule, as we’ll see later.
- RTP and Hit Frequency values are calculated as if a player received every possible combination of reel stops one time, or a full cycle.

The full Liberty Bell spreadsheet is available for review. Feel free to make a copy and play around with it to update the Liberty Bell math however you like. For example, what changes could you make to you increase the RTP to 90% without changing the reel strips? How could you increase the Hit Frequency to 29% or more?

In future posts I’ll present more complicated examples and show how to handle features like multi-line games, Wild symbols. and Scatter symbols. Leave a comment if there are other features you’d like to see illustrated. I’m also happy to hear from other slot game designers and learn whether these concepts match the PAR sheets you’ve created before.

Slot Machines Aren’t Real Games — Or Are They?

In the world of video games, slot machines get a bad rap.

I have worked at game studios where the slot game developers were biding their time, just getting their job done until they could find a way to work on “real games” instead of slots.

Before I started learning how to make good slot games, I shared their concern. At first glance, slots seem like simple, mindless entertainment. How can you possibly make a game compelling and fun to play when the players have so little freedom of choice?

And therein lies the challenge that drives most slot machine designers.

Slot machines are a fascinating blend of art, mathematics, and human psychology.

For a slot machine to be successful, a lot of things need to come together in one package:

A strong and compelling math model
Great pacing and timing
Artwork that “pops”
Catchy music and sounds
A metaphor or narrative that ties the game together

In this age of “analytics paralytics” there is still an art to designing a great slot machine. You can nail all the elements listed above and your game still might not be a hit. There is a final, elusive element that’s needed — dare we call it luck? — that just makes people want to play the game.

I hope to touch on all these elements on this site at some point, though I’ll start first with what I consider the core of a good slot game: the math model.

A slot machine’s math model might not be the biggest attractor for a game — beautiful art, animation and sounds carry greater weight in that regard. But a dull or broken math model can kill a game’s popularity, and when that happens even the most awe-inspiring graphics can’t save it.

This blog will also cover other topics, like pacing and timing, and how metaphor and narrative can make slot games more appealing. I’ll try to demystify some of the closely-held secrets of slot game design, and maybe in the process, I can help add a bit of glitz to an underappreciated form of gaming.

(Photo by Chuma A on Unsplash)