A classic slot designed math-first. Every number below is derived exactly from the reel strips and paytable, including a player-choice respin feature whose return depends on strategy, solved with a Markov chain and checked by simulation.
Lines pay left to right from reel 1. Gears pay nothing, but each gear on screen charges the meter. A full meter lets you respin any reels you choose.
Tap the reels you want to respin. The others stay put. Only the final screen pays, so a respin gives up the current win.
That threshold is the optimal strategy. Spending at the first chance is legal but worth only half as much, which is why this game has an RTP range instead of a single number.
Three inputs define the base game: the reel strips, the ten paylines and the paytable. Base RTP is 83.67%; the feature adds the rest.
Pays in line bets. The last column is each symbol's exact contribution to base RTP, computed from strip frequencies.
| Symbol | ×3 | ×4 | ×5 | RTP share |
|---|
Every cell of the window shows each strip position with probability 1/L, because the stop is uniform and the strip wraps around. So all ten lines share the same distribution, and by linearity of expectation RTP of the game = RTP of one line, however the lines overlap.
Hit frequency cannot come from a formula: lines share cells, so their wins are correlated. These figures come from a full enumeration of all 4,686,066 stop combinations, rerun live in your browser.
Running the exact enumeration…
Rows numbered top 0, middle 1, bottom 2. Lines 1, 8, 9 and 10 all start in the middle cell of reel 1, which is why line wins move together.
One gear per strip, so a reel window holds at most one. Reels 4 and 5 carry extra cherries and lemons, which makes long combinations rarer.
| Symbol | R1 | R2 | R3 | R4 | R5 |
|---|
The feature adds 12.51% RTP under optimal play and 6.64% if the meter is spent at the first chance. Solving it takes three steps.
For a given window and a chosen set of reels R, a held reel keeps its symbol and a respun reel draws from its strip again. Per line, replace p with q:
The window's expected win is the sum over ten lines. Lines through the same respun reel are dependent, but expectation is linear, so this is exact. A line's value depends only on its five symbols, so a table of 7⁵ contents × 32 reel sets answers any query with ten lookups. That powers the in-game advisor.
Check: respinning all five reels from any window is worth exactly the base RTP, 8.37 line bets.
States are 0–19 charges plus "full". Each paid spin moves the chain by the number of gears on screen. On a full meter, the player spends if the gain G = best respin − current win clears a threshold g; otherwise the meter stays full and new gears burn. The stationary distribution π gives the feature's return directly:
Waiting costs only the burned charges, about 0.70 per spin. Scanning g shows the optimum at a gain of 3.5× the total bet.
Operators and regulators see the whole range, so the minimum must also be acceptable. Two design rules follow from the persistent meter: each bet level keeps its own meter, otherwise charges collected at a low bet could be spent at a high one; and the meter is capped so that waiting has a price.
| Strategy | Feature RTP | Respin every | Avg gain / respin |
|---|---|---|---|
| Threshold 3.5× | 12.51% | 38.8 spins | 4.85× |
| Spend at once | 6.64% | 28.8 spins | 1.91× |
The simulator plays the full game, meter and respins included, with the chosen strategy and pays the actual outcomes. It shares no code path with the Markov chain. Three runs of 20 million spins averaged 96.17% ± 0.08% against the exact 96.19%.
Dashed line: exact value. Band: 95% interval ±1.96·σ/√n using the sample σ.
The sample mean wobbles by σ/√n. With the feature, σ ≈ 3.21 total bets per spin, so
| Target accuracy ε | Spins needed |
|---|---|
| ±1% | ≈ 396,000 |
| ±0.5% | ≈ 1.6 million |
| ±0.1% | ≈ 39.6 million |
This is why the par sheet relies on exact methods, and why simulation is used as an independent check rather than as the source of the numbers.
Closed-form line probabilities for RTP and symbol shares · full enumeration of 4.69M outcomes for hit frequency, σ and the win distribution · linearity of expectation for respin values · a 21-state Markov chain for the feature under each strategy.
Added a non-paying gear symbol, one per strip, and a 20-charge meter per bet level. Gears diluted the strips and base RTP fell from 96.06% to 80.39%, so plum and lemon pays were retuned. Feature value solved exactly with a Markov chain; optimal spending threshold 3.5× total bet; RTP range 90.31–96.19%.
5×3 reels, 10 lines, six symbols. Strips and pays tuned to 96.06% RTP; hit frequency, σ and max win from full enumeration; in-browser Monte Carlo convergence check.
Variants of the same game at 94%, 96% and 97% RTP built only from strip changes; a volatility comparison between them.