Cosmic Cascade — Cluster Pays, Solved in Closed Form
Five independent draws, and a symbol pays wherever it lands rather than on a line. The return is the binomial sum over each symbol’s three, four and five-of-a-kind probabilities, solved algebraically at load rather than sampled.
Format: Cluster pays · 5 draws · position-free wins · Demonstrates: A return derived from the binomial rather than measured
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The machine
| Draws | Five independent draws from a fifteen-stop pool |
|---|---|
| Win condition | Three or more of a symbol anywhere in the five — position irrelevant |
| Symbol pool | 🪐×5 · ☄️×4 · 🌟×3 · 🌌×2 · 🛸×1 |
| Pays (3 / 4 / 5) | 🪐 — / 2× / 10× · ☄️ — / 5× / 30× · 🌟 3× / 12× / 80× · 🌌 12× / 60× / 500× · 🛸 40× / 300× / 3,000× |
| Basis of the figure | Exact. Closed-form binomial sum, computed at load |
Position-free wins make the algebra tractable
Paying a symbol wherever it lands removes the geometry from the problem, and removing the geometry is what lets the return be written down instead of estimated. Each symbol’s contribution is its binomial probability of appearing three, four or five times in five draws, multiplied by the pay at that count, and the total is the sum across all five symbols.
That is a formula, not a simulation. The page evaluates it on load and prints the result.
The two lowest symbols pay nothing at three of a kind
🪐 and ☄️ are the common symbols, and hitting three of either returns zero. That is visible in the paytable above rather than discovered after forty spins, and it is a deliberate piece of the design: the frequent near-structure of this game produces three-of-a-kind constantly, and most of those are not wins.
A game that celebrated them anyway would be running the presentation pattern this platform was built to demonstrate against. Under the default presentation mode, a return of zero looks like a return of zero.
The mathematics
Basis: exact. Computed at page load, not sampled — there is no error bar because there is no estimate.
Five independent draws, and a symbol pays wherever it lands — position is irrelevant, so
there are no paylines to enumerate. The chance of drawing exactly k of a symbol in five
draws is the binomial, which means this return is solved in closed form rather than simulated.
| Symbol | Stops | Probability | Pays | Contributes |
|---|---|---|---|---|
| 🌌 ×3 | 2/15 | 0.017804 | 12× | 21.3649% |
| 🌟 ×3 | 3/15 | 0.051200 | 3× | 15.36% |
| 🛸 ×3 | 1/15 | 0.002581 | 40× | 10.3243% |
| ☄️ ×4 | 4/15 | 0.018542 | 5× | 9.2708% |
| 🪐 ×4 | 5/15 | 0.041152 | 2× | 8.2305% |
| 🌌 ×4 | 2/15 | 0.001370 | 60× | 8.2173% |
| 🌟 ×4 | 3/15 | 0.006400 | 12× | 7.68% |
| 🪐 ×5 | 5/15 | 0.004115 | 10× | 4.1152% |
| ☄️ ×5 | 4/15 | 0.001348 | 30× | 4.0454% |
| 🛸 ×4 | 1/15 | 9.22e-5 | 300× | 2.7654% |
| 🌟 ×5 | 3/15 | 0.000320 | 80× | 2.56% |
| 🌌 ×5 | 2/15 | 4.21e-5 | 500× | 2.107% |
| 🛸 ×5 | 1/15 | 1.32e-6 | 3000× | 0.3951% |
| Return to player | 96.4359% | |||
The two commonest symbols pay nothing at three of a kind. That is the design: the
🌌 three-of-a-kind alone carries 21.3649% of the whole
return, and the 3,000× five-of-a-kind on the rarest symbol carries
0.3951%. The headline multiplier is the smallest line
in the table.
Thirteen paying combinations, each one probability times payout, summing to 96.4359%.
What the gate does before the game loads
Nine states have enacted statutory bans on the sweepstakes casino model, and six more closed their markets by enforcement without passing anything. Every one of those instruments turns on redeemable value. This platform has none, so it stays lawful in all fifty — but the geo-gate still names the statute, because a compliance claim that cannot be checked against a citation is a marketing claim.
Select New Jersey and the gate blocks the lobby citing A5447/S4282. Select Oklahoma and it plays, with an amber banner recording that SB 1589 takes effect on 1 November 2026. The distinction between an enacted ban and a ban in force is the sort of thing operators discover late; here it is in the interface, fed by the same weekly-verified database as the state tracker.