The three inputs
A slot's mathematics is fully determined by three things: what symbols sit on each reel and how often, what each winning combination pays, and the rules governing how combinations are evaluated. Everything else, including the RTP, is an output.
Reel strips are ordered lists of symbols, one per reel. A five-reel game has five strips, and each may be a different length with a different composition. A symbol appearing eight times on a 60-position strip has an 8/60 chance of landing in a given position on that reel, provided every stop is equally likely. Some games instead map the random number to stops through a weighted table, as the virtual reels of mechanical reel slots do, and then the weights, not the visible strip, set the probability.
The paytable states what each combination pays, usually as a multiple of the line stake or the total stake. The distinction matters and is a frequent source of confusion when reading a spec.
The evaluation rules determine what counts as a win: the payline structure, whether combinations read left to right or both ways, how wilds substitute, what triggers a feature, and how feature outcomes are computed.
Given those three, the RTP is not a design choice. It is a calculation, and the designer's job is to arrange the three inputs until the calculation produces the target.
How RTP is actually computed
For a simple base game, the method is exhaustive enumeration.
Enumerate every combination of symbol positions the reels can produce. For a five-reel game with strips of 60 positions each, the number of combinations is the product of the reel lengths: 60 to the power of five, or 777,600,000. For each combination, evaluate the paytable and record the total payout. Sum payout multiplied by probability across all combinations, and divide by the stake.
That is computationally trivial for a modern machine and is exactly what maths software does. The important conceptual point is that it is exact, not estimated: every base-game combination is enumerated, so the base-game contribution to RTP is a precise number rather than a simulation result.
Features complicate this. A free-spins round with retriggers has, in principle, unbounded depth, and a cascading mechanic has variable length. Where a feature cannot be enumerated in closed form, its contribution is established by Monte Carlo simulation: run the feature many millions of times, take the mean return, and attach a confidence interval. Testing standards accept both routes: GLI defines a theoretical return to player as one based on mathematical calculations or simulations. Many models use a hybrid, enumerating what can be enumerated and simulating the rest.
Weighting: the designer's actual lever
Changing the paytable changes the game's identity. Changing symbol frequency does not, and that is where most tuning happens.
Suppose a target RTP of 96% and a model currently computing 97.3%. The designer needs to remove 1.3 percentage points. The options, in roughly increasing order of how much they change the feel of the game:
Adjust high-symbol frequency on one reel. Removing a single instance of the top-paying symbol from reel three reduces the frequency of the largest base-game combinations. Small, precise, and almost invisible to the player.
Adjust scatter frequency. Because the feature usually contributes a large share of RTP, scatter frequency is a powerful lever. Reducing it from, say, one in 180 spins to one in 200 removes a meaningful amount of return, and it also changes how often the player reaches the part of the game they are playing for. This is powerful and blunt.
Adjust wild frequency or wild behaviour. Affects both base and feature.
Adjust the paytable. Changes the advertised product.
The general principle is that the lowest-visibility lever that achieves the target is preferred, because a maths model is usually being tuned to hit a commercial requirement after the design has been signed off.
What the par sheet contains
The par sheet, also called the maths sheet or PAR sheet (PAR is usually expanded as probability accounting report, and sometimes as paytable and reel strips), is the internal specification that records all of this.
A complete one carries: the reel strips for base game and any feature reels; the symbol frequency table per reel; the paytable; the payline or ways structure; the computed RTP with its split between base game and each feature; hit frequency, usually both overall and above stake; the volatility index or standard deviation; the maximum win and the probability of reaching it; and the feature trigger frequency. GLI's standards define a PAR sheet more narrowly, as a specification giving the theoretical return to player, hit frequency, symbol combinations, number of reels and reel strip listing; a modern video slot's sheet carries much more.
It is shared with testing laboratories and regulators and is not normally published, although GLI's standard for online systems expects the operator to keep documentation such as PAR sheets recording each game's theoretical RTP, and researchers in Ontario have obtained PAR sheets through freedom-of-information requests. What reaches the player is the RTP, since in Britain the Gambling Commission requires the house edge, the RTP or the likelihood of winning to be easily available before the customer commits to gamble, often a volatility rating and sometimes hit frequency, which is why a maths-literate operator asking a studio for the full figures is asking for something the studio can supply and usually does not volunteer.
Base game versus feature split
One number on the par sheet explains more about a game than any other: how the RTP divides between base game and features.
A game at 96% RTP might split 60/36, with 60 percentage points from base-game wins and 36 from the free-spins round. Another 96% game might split 88/8.
The first is a feature-led game. The player's return depends heavily on reaching the feature, the base game is close to a waiting room, and the experience is dominated by anticipation. The second is a base-led game with a modest bonus, which plays as a steadier grind.
This single ratio predicts most of what a player will say about a game, it determines how bonus-buy pricing works, and it is the number most often omitted from a commercial spec sheet. Ask for it.
A worked miniature
The figures below are deliberately tiny and round so the arithmetic is legible. A real model is the same process across a vastly larger space.
Consider a one-reel, one-payline game. The reel has 20 positions: 1 diamond, 3 bells, 6 cherries and 10 blanks. The paytable pays 30 for a diamond, 6 for a bell, 2 for a cherry, and nothing for a blank. Stake is 1.
- Diamond: probability 1/20 = 0.05, payout 30, contribution 1.50
- Bell: probability 3/20 = 0.15, payout 6, contribution 0.90
- Cherry: probability 6/20 = 0.30, payout 2, contribution 0.60
- Blank: probability 10/20 = 0.50, payout 0, contribution 0.00
Expected return = 1.50 + 0.90 + 0.60 = 3.00. That is an RTP of 300%: a game that on average pays out three units for every unit staked, and so loses the operator money over any meaningful run of play, even though half its spins land on a blank and pay nothing.
To bring it to 96%, the designer has to remove 2.04 of expected return. Replacing the diamond with a blank, so the reel stays at 20 positions, removes 1.50, leaving 1.50 or 150%. Reducing cherries from 6 to 2, again replacing them with blanks, removes a further 0.40, leaving 1.10. Reducing bells from 3 to 1 removes 0.60, leaving 0.50 or 50%, which has overshot.
The point of the exercise is not the arithmetic. It is that symbol frequency and payout are in direct tension, that the adjustments interact, and that hitting a precise target requires iteration rather than a single calculation. Now imagine it across five reels, 60 positions each, 25 paylines, wilds that substitute, and a feature contributing a third of the return, and you have a sense of why maths design is a specialism.
Why a certified game cannot adjust to the player
The model above is fixed at design time and frozen at certification. In Britain, a game's rules, payouts and outcome probabilities may not be changed while it is available for gambling, except as the game's own rules provide. The random number generator selects positions from the reel strips; it does not know who is playing, what they have staked, or whether they are ahead.
This is worth being able to explain clearly, because it is a common belief among players, close to the belief that some machines are "hot" or "ready to pay out" which researchers have documented, and in certified markets it is false. There is no mechanism in a certified slot for the outcome distribution to respond to a player's history: in Britain adaptive behaviour, that is a compensated game, is not permitted, GLI's standard states that a game shall not adapt its theoretical return to the player based on past payouts, and certification exists specifically to verify that.
Two genuine phenomena get mistaken for it. Games with multiple certified RTP configurations can differ between operators: Ontario, for example, approved versions of one slot with payback percentages from 85% to 96.2% that looked identical to the player, which is why those researchers argue that loose and tight machines, in this sense, do exist. This is legitimate where disclosed and is a per-integration setting rather than a per-player one. And a player on a high-volatility game who has had a long losing run is experiencing exactly what the model predicts, which feels like the game tightening and is not.