Cg11: Tournament model and Round Arrow Triangles

Every unordered pair of five labelled vertices independently chooses one of its two directions with equal probability. All ten pairs therefore have exactly one arrow, including switches encoded0. The raw score counts the ten three-vertex subsets whose arrows form a directed cycle, such as0→1→2→0. Transitive triangles score nothing. Each qualifying triple counts once, independent of where its description starts. The software selects directions and counts cycles automatically.

If every arrow points from a lower label to a higher label, the tournament has no directed cycle and scores0. For a five-vertex cyclic orientation in which vertex i points to i+1 and i+2 modulo5, five triples form cycles and the raw score is5. Although there are ten triples, five is the maximum score for this model. Arrowheads display actual direction; an encodedzero does not mean an absent edge.

Choose a stake and weighting before pressing Draw. One whole stake is debited once for the entire graph. There are no separate rows, paylines or row shares. Settings and refill remain locked while the draw settles and throughout an active automatic queue. Auto schedules up to ten rounds, subject to sufficient balance. Stop cancels future rounds; it does not refund or change an already drawn result.

If every arrow points from a lower label to a higher label, the tournament has no directed cycle and scores0. For a five-vertex cyclic orientation in which vertex i points to i+1 and i+2 modulo5, five triples form cycles and the raw score is5. Although there are ten triples, five is the maximum score for this model. Arrowheads display actual direction; an encodedzero does not mean an absent edge.

The ten fair binary direction choices give1024 equally likely tournaments. Counts for scores0,1,2,3,4,5 are120,120,240,240,280,24. Exactly904 outcomes score positive; they need not be profitable after subtracting the stake. The three weighting modes preserve these positive outcomes and their frequency. Previous tournaments do not affect the next draw.

Before a draw choose exponent p=1,2 or3. Gross multiplier=0.96 × score^p / mean(score^p). Exact means are2.5,8.125 and28.75. Each mode therefore returns expected gross credit96% of its captured stake. This is not a96% chance of winning or a guaranteed wallet result. A zero score stays zero; integer credit is settled with saved fractional carry.

Choose a stake and weighting before pressing Draw. One whole stake is debited once for the entire graph. There are no separate rows, paylines or row shares. Settings and refill remain locked while the draw settles and throughout an active automatic queue. Auto schedules up to ten rounds, subject to sufficient balance. Stop cancels future rounds; it does not refund or change an already drawn result.

Enumerating the1024 tournaments and checking the formula can verify the Round Arrow Triangles model. That does not establish an outside catalogue, certify commercial randomness, confirm a licence or demonstrate another service’s safety. The simulator has no blackjack strategy, crash cashout or sports-prediction feature. It provides no external audit or endorsement.

A new browser wallet starts at 1000 virtual coins. Settled integer balance and fractional payout carry are saved in localStorage. A gross credit is added to any saved fraction; the integer part is credited and the remaining fraction is carried forward. These coins have no cash value, deposits, withdrawals or prizes. Storage belongs to this browser and origin, not to an account or an external payment balance.

Only an idle balance below ten offers Free refill. Clicking it sets the balance to 1000; it does not add 1000 to the existing balance. The fractional carry is preserved. Refill is unavailable during settlement or an active automatic queue. Changing the stake cannot force a refill while the balance is ten or more. A previously saved zero balance remains zero after reload until the user clicks the available refill.

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