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
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.
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.
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 highestpaying matching combination. In this example, WildWildWild would be treated as BellBellBell and pay 20.
The formula for the BellBellBell 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 higherpaying WildWildWild 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.
My advice for firsttime slot designers is to avoid putting Wilds on the first reel, at least until you are comfortable using simpler Wildsubstitution 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.