How the King Closed Tour Board Works
The King Closed Tour demo uses a 4 by 4 board, containing 16 cells identified in row-major order from 1 to 16. Before the round begins, a fixed number of hazards are hidden uniformly among all complete layouts that satisfy a specific constraint. This constraint requires that the eight-neighbour king hazard graph has a Hamiltonian cycle visiting each hazard once and returning to its start, with at least three hazards required to form a valid layout.
King Closed Tour samples one complete 4×4 hazard layout uniformly from the published constraint support of the selected count. This layout stays fixed for the duration of the round. When you pick a cell, it reveals whether it is safe or a hazard based on this pre-determined layout. The hidden layout is never redrawn during the round.
Exact Scoring and Probability Math
Each selected safe cell is priced from the exact posterior over consistent layouts. The gross value starts at 0.97 times your stake and becomes the current value divided by that cell's safe probability. Known-safe cells leave the value unchanged, while proven hazards are disabled from selection. The live gross quote is calculated as 0.97 times the original support count divided by the surviving support count.
After safe picks, only compatible layouts remain, and each cell shows its exact conditional safety probability. Already revealed cells, proven safe cells, and proven hazards cannot be selected. A hazard returns zero. A choice with a prospective quote above 1,000 times the original stake is blocked without changing state. Every revealed-information strategy has a 97% expected gross return in the fair support model, which is a mathematical expectation and not a win probability.
Settlement and Collection Rules
Collection is allowed only after at least one safe reveal. Individual prospective values above 1,000 times the original stake are disabled. Automatic settlement occurs only when no eligible uncertain move remains. If no uncertain cell remains after a safe pick, settlement is automatic. A choice with a prospective quote above 1,000x is blocked before revealing, ensuring no state change occurs during the block.
There is no automatic queue. You may collect after the first safe pick. The actual payout math records every mode, and displayed multipliers represent gross coin returns, not profit. The game caps and resets are mechanics, not guarantees of safe behavior.
Virtual Wallet and Refill Mechanics
The demo uses a local virtual wallet with an initial balance of 1,000 coins. The fractional carry persists through settlement, manual refill, and reload. The settled balance includes zero and persists locally. An unfinished round ends without refund or resume; reloading the page does not restore unfinished progress. A valid saved balance is loaded, including zero; 1,000 coins is used only when no valid saved balance is available.
Idle balance below 10 allows a manual restoration to 1,000 coins, preserving any fractional change. This restoration is not automatic and does not add the reset amount again. Settings and refill are locked while active. One whole stake of 10, 20, 50, or 100 is captured once at round start. Sound remains available during play.
Controls, Modes, and Locking
The controls are locked during active play to ensure the fixed layout remains consistent. Stake, mode, and refill are locked while active. Sound and eligible uncertain picks remain available. Collect becomes available after at least one safe reveal. You cannot change settings or refill during an active round.
The interface distinguishes between active play modes and mathematical simulation strategies. The eight-neighbour king hazard graph defines the constraints, but the user interface simply presents the 4x4 grid with conditional probabilities. The exact modes and controls are tied to the fixed layout sampling.
Mathematical Evidence and Simulation
Mathematical evidence shows that for depths 1, 2, and 3 with hazards 3, 4, and 5, the exact expected return is consistently around 97%. For example, with 3 hazards and a support count of 36, the exact percent is 97.00000000000001. With 4 hazards and a support count of 61, the exact percent is 97.00000000000006. With 5 hazards and a support count of 116, the exact percent is 97.00000000000003.
Measured simulation percentages from 1,000,000 rounds align closely with these exact values, such as 96.998975% for depth 1 with 3 hazards. These are sample observations and not exact rules. The expected return is a mathematical expectation, not a promise of win probability over one session.
Frequently asked questions
Does the hazard layout change after each pick?
No. One complete 4×4 hazard layout is sampled uniformly from the published constraint support of the selected count and stays fixed for the entire round. The hidden layout is never redrawn.
What is the expected return of King Closed Tour?
Every revealed-information strategy has a 97% expected gross return in the fair support model. This is a mathematical expectation, not a win probability or a guarantee over one session.
When can I collect my winnings?
Collection is allowed after at least one safe reveal. You may collect after the first safe pick. Automatic settlement occurs only when no eligible uncertain move remains.
What happens if my balance drops below 10?
If your idle balance is below 10, you can manually restore it to 1,000 coins. This preserves any fractional change. The restoration is not automatic and does not add the reset amount again.
Do settings and refill lock during play?
Yes. Stake, mode, and refill are locked while active. Sound remains available during play.
What is a prospective quote above 1,000x?
A choice with a prospective quote above 1,000 times the original stake is blocked without changing state. This prevents selection when the value exceeds this cap.
Does an unfinished reload refund my stake?
No. An unfinished round ends without refund or resume. Reloading the page does not restore unfinished progress.
How are hazards distributed on the board?
Hazards are hidden uniformly among complete layouts satisfying the rule that the eight-neighbour king hazard graph has a Hamiltonian cycle visiting each hazard once and returning to its start, with at least three hazards required.
Are displayed multipliers net or gross returns?
Displayed multipliers are gross coin returns, not profit. The gross value starts at 0.97 times your stake.
Is the virtual wallet linked to an account?
No. The demo uses a local virtual wallet. Game source uses localStorage for settled balance and fractional carry. There is no account flow or payment flow in the demo.
What is the support count?
The support count is the number of complete layouts satisfying the constraint for a given hazard count. For example, with 3 hazards, the support count is 36. With 4 hazards, it is 61. With 5 hazards, it is 116.
Does the game track my personal data?
The demo stores settled balance and fractional carry in localStorage. This does not establish anonymity or exclude other processing. The published page includes Google Fonts, Statinside counter.js with ID 44 and a noscript image at statinside.com/pixel.png?id=44. Only that image has referrerpolicy=unsafe-url. These declarations do not establish complete third-party processing or retention practices.
