How Cyclic Bridges counts starting positions

Each ball has twelve independent fair choices. Left contributes −1 and right contributes +1. The game tests all twelve labelled starting positions around the circular word. A position qualifies when its rotated word never has a negative running sum and finishes at zero. Repeated rotation values still count separately because their starting positions are different. The score is the number of qualifying positions.

All-left and all-right words score zero: their final sums are −12 and +12. An alternating left/right word has six qualifying positions, all starting on right. A word with six consecutive right choices followed by six left choices has one qualifying position. Any word with unequal counts of left and right scores zero.

Each ball has twelve independent fair choices. Left contributes −1 and right contributes +1. The game tests all twelve labelled starting positions around the circular word. A position qualifies when its rotated word never has a negative running sum and finishes at zero. Repeated rotation values still count separately because their starting positions are different. The score is the number of qualifying positions.

All-left and all-right words score zero: their final sums are −12 and +12. An alternating left/right word has six qualifying positions, all starting on right. A word with six consecutive right choices followed by six left choices has one qualifying position. Any word with unequal counts of left and right scores zero.

Select power 1, 2 or 3. The gross return is 0.96 times the ball’s captured whole stake times its raw score raised to that power, divided by that power’s own exact mean. The means are 0.38671875, 0.8701171875 and 2.4814453125, respectively. Each is calculated over all 4,096 equally likely words. The three modes preserve the same zero-score results but change the size of positive returns. Expected gross return is 96% of stake; this is not a promise of net profit.

Up to six balls can be active at once. Each ball captures its whole stake and selected power at launch, and the stake is debited then. Changing settings affects later launches. In normal motion, the animation runs for about 1,200 ms; reduced-motion preference can settle it earlier. There is no manual early collection. Auto schedules at most ten launches; Stop cancels future launches while existing balls finish. Sound remains available.

A new wallet starts with 1,000 virtual coins. The current balance, including each stake debit at launch and any zero balance, and fractional carry persist in the browser where storage is available. When all balls have finished, the queue is empty and balance is below 10, manual refill sets the balance to 1,000 while keeping fractional carry. It is locked while any ball or queued launch remains. Reloading discards unfinished balls without refund or resume.

All 4,096 twelve-choice words can be enumerated exactly. Each mode has 96% expected gross return by its own normalization. The one-million-round samples measured approximately 96.04%, 95.95% and 95.70%; these sample observations vary and do not replace the exact probability law. A positive gross return can still be smaller than the stake.