The private urn
Each ball starts with left and right weights of one. For twelve rows, choose uniformly among the total integer weight, follow that side, then increase its weight by one. A new ball resets both weights. Reinforcement applies within a ball, so past completed balls cannot make a new ball owe a particular result.
Exact path probabilities
A path with r right turns has probability r!(12−r)!/13!. The 4,096 binary paths are not equally likely. Summing paths by right count yields thirteen equally likely terminal counts, each 1/13. Thus zero-score probability is 1/13 in all powers. This calculation concerns the implemented model, not an external service.
Score powers and means
Raw score is (2R−12)² where R counts right turns. P1, P2 and P3 use raw, raw² and raw³ with separate exact means 56, 5600 and 661376. Gross reward is 0.96 × stake × powered score / mean. Higher powers change variability, not expected return; six right turns yield zero in every mode.
Reading expected return
Expected gross reward is 96% of the stake; expected net change is therefore −4%. Gross is not profit and a session result is not guaranteed. An all-left P1 ball with stake ten has raw 144 and gross 24.685714… coins. Subtract the ten-coin stake to interpret its net result.
Launch and stop
Each valid launch immediately spends one whole stake and captures its mode. Up to six balls can run together, settling after 1.2 seconds. Changes affect later balls only. Auto schedules at most ten launches; Stop cancels future launches while active balls settle. There is no early cashout or independent stake per row.
Local balance and carry
The wallet begins with 1,000 virtual coins. Exact fractional rewards accumulate as carry, and only whole coins join the spendable balance. When fully idle and below ten, manual refill sets the balance to 1,000 and preserves carry. Refill stays locked while any ball or queued launch remains. Coins have no cash value.
