Core Mechanics and Board Layout
The game is played on a 4×4 board with printed labels 1 to 16. Before a round begins, the player selects the number of hazards: 3, 4, or 5. The selection of hazard labels is not uniform across all combinations; instead, the game uniformly samples only from the complete subsets of labels that satisfy the condition: The product of the selected printed labels has exactly two distinct prime factors.
These specific subsets form the finite support for the game. All safe observations filter this support. Crucially, the printed labels on the board are not clues to a hidden independently uniform grid; they are fixed identifiers used to calculate the product condition.
Only distinct prime factors count: labels 4, 8 and 9 have product 288, whose distinct prime factors are 2 and 3, so this three-hazard layout is allowed. Labels 2, 3 and 5 have three distinct prime factors and are excluded.
Pricing and Probability
Start with a gross value of 0.97 × the captured stake. Before a reveal, count the currently compatible layouts in which the chosen cell is safe and divide by all currently compatible layouts. That fraction is its exact conditional safe probability p.
If the reveal is safe, the new gross value is the current gross value divided by p. A probability-1 reveal leaves the value unchanged. A hazard ends the round with zero return. Filtering the layouts after each safe reveal changes later probabilities.
Controls, Modes and Stakes
Players can select stakes of 10, 20, 50, or 100 play coins, with 20 being the default. Settings, restoration, and starting a new round are locked while a round is active. The player chooses 3, 4, or 5 hazards before the round starts.
- Collection: Players can collect only after at least one safe reveal has occurred.
- Auto-Bank: The game automatically banks only when no eligible uncertain choice remains.
- Disabling Choices: Individual choices where the gross return exceeds 1000× the captured stake are disabled, while equality (exactly 1000×) is allowed.
Settlement and Wallet Persistence
At settlement, the floor of the raw gross return plus the stored fractional carry is credited to the wallet; any remainder persists in the fractional carry. Rounds with zero return leave the fractional carry unchanged.
The wallet state is persistent in the browser's local storage. The initial balance is 1000 virtual coins. Restoration is manual and idle-only: if the balance drops below 10 while idle, the user can reset it to exactly 1000. Refilling coins does not clear the fractional carry.
If a round is unfinished when the page reloads, the captured stake remains deducted. The round ends without refund, payout, or continuation. Controls reset to the default stake of 20 and the first preset upon reload.
Mathematical Evidence and RTP
The expected RTP is 97% across all modes. Simulation measurements are supporting observations and not the mathematical payout law. The table contains saved simulations of 1,000,000 rounds for each listed hazard-count and stopping-policy combination. The measured RTP and standard error are sample results, not exact payout constants. Each mode’s theoretical gross return is 97%, not guaranteed session profit.
The saved policy names describe stopping tests: low-1 chooses an eligible cell with the lowest current safe probability and collects after one safe reveal; high-2 chooses the highest current safe probability and collects after two safe reveals; first-4 chooses the first eligible printed label in ascending order and collects after four safe reveals. A hazard returns zero, and automatic banking can finish a round before the requested reveal count.
| Hazards | Policy | Support Count | Expected RTP (%) | Measured RTP (%) | Std Error % |
|---|---|---|---|---|---|
| 3 | low-1 | 167 | 97 | 97.00313352282944 | 0.01871772765784715 |
| 3 | high-2 | 167 | 97 | 97.01755550824032 | 0.12151486054491191 |
| 3 | first-4 | 167 | 97 | 96.68334580968819 | 0.2554056029395452 |
| 4 | low-1 | 209 | 97 | 97.00037144021773 | 0.013548244050409056 |
| 4 | high-2 | 209 | 97 | 96.946753062365 | 0.17761529261661513 |
| 4 | first-4 | 209 | 97 | 97.8780439999262 | 0.5233437555797064 |
| 5 | low-1 | 176 | 97 | 97.00564041355621 | 0.0072953196295048334 |
| 5 | high-2 | 176 | 97 | 96.57853078279061 | 0.24971794415289658 |
| 5 | first-4 | 176 | 97 | 98.38593599998836 | 1.2922712203671805 |
Worked Example
Before any reveal in a 3-hazard round, 167 layouts are possible. Label 4 is safe in 119 of them, so its initial safe probability is 119/167 ≈ 0.7125748503. At stake 20, the initial gross value is 19.4. If label 4 is revealed safely first, the gross value becomes 19.4 ÷ (119/167) ≈ 27.2252100840.
With no earlier fractional carry, collecting then credits 27 coins and stores about 0.2252100840 as carry. The stake was already deducted: the gross credit is not net profit. If label 4 is a hazard, the return is zero and existing carry is preserved.
Frequently asked questions
How are hazards selected in Two Prime Product Mines?
Hazards are selected by uniformly sampling from complete subsets where the product of the printed labels (1-16) has exactly two distinct prime factors. This restricts the allowed layouts; sampling within that restricted set is still random.
What happens to my balance if I reload during an unfinished round?
The captured stake remains deducted. The unfinished round ends without refund, payout, or continuation. Controls reset to the default stake of 20 and the first preset.
Is there an automatic account recovery or withdrawal?
No. The demo has no account-login flow or payment flow. Restoration is manual and idle-only, setting the balance to exactly 1000 if it drops below 10. Fractional carry persists in local storage but does not imply external account recovery.
