A board with constrained layouts
The board has four rows and four columns, labelled 1–16 from left to right and top to bottom. Before starting, choose 3, 4 or 5 hazards. One complete layout is drawn uniformly from those satisfying this rule: the sum of products of labels in every distinct hazard pair must be prime. Placement is random within that allowed set, rather than independent equally risky hazards. There are 128, 458 and 990 layouts for 3, 4 and 5 hazards.
Multiply each pair, then add
For three hazards a, b and c, the sum is ab+ac+bc, not a+b+a+c+b+c. Count each pair once without pairing a label with itself. Four hazards give six pairs; five give ten. A prime number is greater than one and has only positive divisors one and itself. Layout {1,4,5} is allowed: 1×4+1×5+4×5=29, which is prime. Individual labels need not be prime: one is not prime and four is composite. Layout {1,2,4} is not allowed because its product sum is 14.
Probability after safe openings
Each safe opening removes layouts that would put a hazard there. For an unopened cell, safety probability is the number of remaining layouts in which it is safe divided by the total remaining layouts. The game uses that exact fraction. A button percentage is rounded to one decimal place and is not the full value used in pricing. Probabilities differ between cells and change with observations. A zero-probability cell is blocked; a probability-one cell is safe and does not increase gross value.
Divide current value by safety probability
Starting deducts and saves one stake. Initial gross value is 0.97 × stake. If the selected cell is safe, divide current gross value by its exact safety probability before opening. At probability one, value is unchanged. Finding a hazard makes the round’s gross return zero and retains previous carry. The displayed value is rounded to three decimals; settlement uses the full value. Opening a cell does not credit the wallet before collection or automatic settlement.
Collect after one safe opening
Collect becomes available only after at least one safe opening. You may collect current gross value or choose another eligible cell. A cell choice is blocked if current value divided by its safety probability exceeds 1,000 times the stake; equality is allowed. This limit applies to choosing a cell, not to Collect. After a safe opening, automatic collection occurs only when no eligible unopened cell has uncertain safety; certainly safe cells may still remain unopened.
Exact expectation and finite samples
Independent enumeration of layouts and probabilities confirms 97/100 expected gross return per stake for completion policies respecting eligible choices and collection. Nine policies were checked: choose the lowest eligible label once; the highest up to twice; or the lowest-safety-probability cell up to four times, stopping on a hazard or automatic collection. One million draws per policy gave finite observations around the expectation, with differing variance. Expectation is not a win probability, guaranteed profit or session forecast. Closing before collection forfeits pending value.
Frequently asked questions
How are hazards drawn?
Uniformly among 3-, 4- or 5-hazard layouts whose sum of products of all distinct label pairs is prime.
Does 100% safety increase return?
No. Dividing by probability one leaves gross value unchanged; collection is available after that opening.
Does the 1,000-times limit block Collect?
No. It limits cell choices by potential return after a safe opening. Collect requires at least one safe opening.
