Euclidean Steps FAQ: Ordered Pairs and Saved Coins

Draw two independent uniform integers from one through sixteen, retaining their drawn order. While the second number b is nonzero, replace the pair (a,b) by (b, a modulo b) and count one division. Count BOTH an initial division with quotient zero when a is smaller than b and the final division whose remainder is zero. Do not sort the pair. Raw score is max(0, division count minus two), not the final greatest common divisor.

For (1,2), the divisions are 1 divided by 2 with remainder 1, then 2 divided by 1 with remainder 0. There are two steps and score zero. (2,1) has only one step and also score zero. For (2,3), remainders are 2, 1 and 0, giving three steps and score one. The initial quotient-zero division counts even though the first number did not decrease.

Select power one, two or three before drawing. Gross multiplier is 0.96 times raw score raised to that power divided by the matching exact unconditional mean. Those means are 0.7421875, 1.3515625 and 2.921875. Zero stays zero. Every mode has 96% exact unrounded expected gross credit of captured stake, not winning frequency or a session promise. Over all 256 ordered pairs, 130 score zero, 75 score one, 39 score two, 11 score three and one scores four. A positive score can still return less than stake.

Choose a whole virtual stake and power before starting. One complete captured stake is debited once. The sampled pair settles after animation, normally 480 milliseconds; reduced motion changes duration rather than probabilities. Settings and refill lock during settlement and an active Auto queue. Auto schedules at most ten rounds. Stop cancels future queued rounds while the active round settles. There is no early cashout. Wait until idle before changing settings or refilling.

The wallet starts with 1000 free virtual coins. Settled wallet and fractional-credit carry persist in browser localStorage. Only while idle below ten coins can manual refill set balance to 1000 preserving carry; it does not add 1000. Saved debited wallet and carry survive reload, but an unfinished round is not recovered or refunded. Keep an active round open until settlement. No account synchronizes devices or restores removed values.