Adding remainders over divisors
Draw independent numbers a and b from 0 to 12, each equally likely. List every positive divisor of b+1. Divide a+1 by each divisor and add the integer remainders. That sum is the score. Divisor 1 always contributes zero. Each divisor and its remainder counts once, including b+1 itself; this is neither a sum of divisors nor just a count of them.
Three score tickets
Lower accepts {0,1,2,3,4}, covering 115 of 169 pairs. Middle accepts {5,6,7,8,9,10}, covering 43 pairs. Upper accepts {11,12,13,14,17,22}, covering 11 pairs. Values 15,16,18,19,20 and 21 cannot occur as scores in this demonstration. These are bands of computed results, not sectors of a conventional roulette wheel. Select one ticket before drawing; each possible score belongs to exactly one ticket.
Example: adding four remainders
For [4,5], a+1=5 and b+1=6. Positive divisors of 6 are 1,2,3 and 6; remainders of 5 are respectively 0,1,2 and 5. Their sum is 8 and hits Middle. With stake 20, starting wallet 1,000 and no old carry, raw gross return is approximately 75.460465: credit 75, carry 0.460465 and final wallet 1,055. Net change is +55 coins. Another selected ticket does not pay for that same result.
One stake for two results
Choose stake 10, 20, 50 or 100 virtual coins, default 20, and a score ticket. Starting deducts and saves one stake before revealing the two results. During the round, stake, ticket, restoration and another draw are locked; sound remains available. There is no intermediate collection, pause or adjustment after the first result. After ordinary settlement, the chosen ticket and stake remain selected.
Whole-coin credit and net result
On a hit, raw gross return is stake × ticket multiplier. Add saved carry, credit the whole part and retain the new fraction below one coin. On a miss, gross return is zero and old carry remains unchanged. Net whole-coin change is the credit minus the stake already deducted; gross return is not profit. The demonstration has one settlement at the end, with no additional payout for each reading.
Reloading forfeits an unfinished round
Saved wallet, including zero, and carry remain in localStorage while the browser retains those data. Reloading during a round discards its progress: the stake stays deducted without refund, payout or continuation, even after the first reading. Controls return to stake 20 and Lower, with no active round. Ordinary completed settlement retains the selected stake and ticket. A local wallet supplies no password recovery.
Frequently asked questions
Can I change the ticket after the first result?
No. Both readings use the same captured stake and ticket. Stake, ticket, restoration and next-draw controls are locked; sound remains available.
How can a zero score pay?
Zero belongs to Lower. If Lower is selected, score zero hits and pays its gross multiplier. A zero computed score is not an automatic miss.
Does reload return an unfinished stake?
No. The stake stays deducted without payout, refund or resume. Saved wallet and carry remain; controls return to stake 20 and Lower.
