Two hexadecimal reels and one integer
Unitary Totient Reels draws two independent uniform values from 0 to 15. The symbols 0–9 represent themselves and A–F represent 10–15. Read the first value as a and the second as b: the integer is n=16a+b+1, uniformly distributed from 1 to 256. For each full prime power p^e in n’s factorization, use p^e−1; multiply those factors. With no prime factors, the product is 1. Subtract 1 to obtain the reward score. This is the unitary totient statistic, not the ordinary Euler totient.
Stakes and reward curves
Choose one stake of 10, 20, 50 or 100 virtual coins; the default is 20. Choose the linear, square or cube curve before spinning. If the score is s, their weights are s, s² and s³. Both reels share the one stake deducted at the start. Stake, curve, restoration and another start lock until settlement; sound remains available. There are no row bets, bonus rounds, automatic repeated spins or mid-spin collection.
Exact expectation and a worked outcome
Averaging each curve’s gross multiplier over all 256 equally likely integers gives exactly 0.96, or 96% expected gross return per coin staked. This is neither the chance of winning nor a promised session result. For symbols 0 and B, n=12; its prime powers are 4 and 3, the statistic is (4−1)(3−1)=6 and the reward score is 5. Linear weight is 5, square weight 25 and cube weight 125. Symbols 0 and 0 give n=1, statistic 1 and score 0, so every curve returns zero.
Gross return and fractional settlement
The exact sums of the 256 weights are 22,945 for linear, 3,024,331 for square and 479,301,235 for cube. Each mean is its sum divided by 256. The gross multiplier is 0.96 × weight / mean. Multiply that multiplier by the captured stake to get the raw gross return. Add existing fractional carry, credit the whole-number part and save the remainder. Net wallet change is credited coins minus the stake deducted earlier. A zero-score result pays zero and leaves carry unchanged.
Local balance and restoration
A fresh wallet begins at 1,000 virtual coins. The settled balance, including zero, and fractional carry are stored in browser localStorage. When no round is running and the balance is below 10, the restore button sets the balance to exactly 1,000; it does not add 1,000 and is not automatic. Restoration preserves carry. Clearing cached files and removing site storage are different operations; removing this site’s localStorage removes these saved values.
Reloading an unfinished spin
Reloading while the reels are settling leaves the captured stake deducted. The unfinished spin receives no refund or payout and cannot be resumed. The saved remaining balance and carry stay local. Controls return to stake 20 and the linear curve. A saved wallet is not saved unfinished progress.
Frequently asked questions
How is the reward score calculated?
Multiply p^e−1 for the full prime powers in the selected integer’s factorization, using 1 for an empty product, then subtract 1. For n=12 the statistic is 6 and score is 5. It is not the ordinary Euler totient.
When can I restore the balance?
Only while no round is running and the balance is below 10. The button sets it to exactly 1,000 coins and preserves fractional carry. This action can be used whenever those conditions are met.
Can I resume a spin after reloading?
No. Its stake remains deducted and it receives no refund or payout. Saved balance and carry remain, while controls reset to stake 20 and the linear curve.
