THE VALUE FIRST. THEN THE TRADEOFFS.
What does “best play” mean?
The best paytable return is a mathematical fact. The best balance between losing less, leaving ahead and chasing a larger win depends on the player’s objective. KenoBro calculates the return first, then compares four explicitly defined play preferences. The bankroll, machine rules, available denominations and selected number of sessions constrain the plans.
The financial benchmark
RTP is expected payout divided by the amount wagered. It is calculated exactly from the paytable, including rare awards. The highest-return spot row is shown independently of the strategy search. A mixture of rows has a wager-weighted average return, so it cannot exceed the highest row under the model’s linear payout assumptions. More cards and overlapping numbers cannot improve that expectation.
If every paying row returns below 100%, keeping the bankroll unbet maximizes expected final wealth. For someone choosing to play, the lowest expected loss depends on both house edge and total turnover: expected loss equals expected amount wagered × (1 − RTP) for a fixed portfolio. A high probability of a small profit can coexist with a negative average return.
The UK Gambling Commission explains RTP and house edge, including why a session can differ from the long-run average and why past wins and losses do not alter a random machine’s next-game odds.
Why cash exits do not beat the house edge
A gain exit, loss floor or break between days changes the distribution of ending balances and the amount wagered. It does not improve the return on the next draw. With a finite number of plays, stopping does not turn a negative-expected-value game into a positive one. See MIT’s notes on martingales and stopping.
Kelly betting optimizes expected logarithmic growth and can incorporate drawdown constraints; it does not create an advantage. For any strictly negative-expected-return bet, the expected log change is below the log of the expected wealth multiplier, which is negative. Retaining cash has zero log change. That is why this calculator does not present a positive Kelly stake for an all-negative table. See Busseti, Ryu and Boyd’s research on growth and drawdown risk. The four play options here are preference comparisons for people choosing to gamble, not a claim to implement that paper’s optimizer.
The four play objectives
Reserved · lower expected cost. Minimizes modeled expected loss among the tested playing policies. Uses a 10–25% total loss allowance, an initial wager ceiling of 1% of bankroll and a payout standard-deviation ceiling of 3%. Only portfolios within three percentage points of the best return are eligible. On a negative-edge table, reducing turnover reduces expected cost, so this option can recommend just one minimum-cost play per visit.
Balanced · chance of finishing ahead. Maximizes the lower end of the 95% interval for finishing the selected period with more than the initial bankroll. Uses a 25–50% loss allowance, a 2% wager ceiling and a 4% payout-standard-deviation ceiling. Only portfolios within 1.5 percentage points of the best return are eligible.
Ambitious · gain compared with risk. Ranks the period’s gain-goal size multiplied by the conservative probability of reaching it, divided by the loss allowance. Equal goal scores favor a higher good-luck ending balance. Uses a 25–50% loss allowance, 3% wager ceiling and 12% payout-standard-deviation ceiling, with a return tolerance of three percentage points. At most half the card marks may be repeats.
Aggressive · larger realistic upside. Maximizes the estimated 90th-percentile final bankroll, rather than claiming to maximize expected wealth. Uses a 50–100% loss allowance and a 5% wager ceiling. Full stacking and lower-return rows remain eligible when they offer a different payout distribution. This can produce much worse median or poor-luck results.
Generated Reserved and Balanced portfolios reject overlaps that have higher exact variance than their matching separated cards at the same spots and stakes. Unusual zero-hit or non-monotonic payouts are evaluated from the entered table. The percentages above are explicit calculator design choices, not universally optimal gambling rules established by the cited research.
One session or repeated sessions
Choose one visit or 1–365 daily sessions. Each simulation starts with one bankroll. Its full ending balance, including all winnings, becomes the next session’s starting balance. No top-ups, resets or withdrawals are invented.
To support the requested visits, the plan reserves loss allowance for the sessions still to come. Before each visit, divide current funds above the plan’s protected balance by visits remaining. That is today’s maximum loss. Unused allowance and winnings carry forward. The gain exit is a strategy-selected multiple of that allowance. A losing first visit cannot consume the allowance reserved for later visits.
The card structure stays fixed. The bet is flat within a session. Future stakes can reduce to legal denomination/credit combinations but never rise above the initial recommendation; there is no loss-chasing progression. The wager must fit the style’s current-bankroll exposure limit and today’s loss allowance. Stop before another wager could cross today’s floor, after the gain goal, or at the automatic play horizon. If no legal bet fits, that visit is recorded without a bet and the remaining money is retained.
Changing the number of sessions can change a feasible bankroll plan and its outcome distribution. It does not change the underlying spot odds or paytable return. The search evaluates plans across the selected period, rather than taking the best single-visit result and calling it optimal for a month.
The “At the machine” guide lets you enter the session number and actual current bankroll. It uses the same stake and exit rules as the simulations and returns today’s denomination, card credits, gain exit and loss floor.
What is searched, and what is proven
Expected return and covariance-based volatility are exact. The search generates all paying spot rows and two-row mixtures for active-card counts considered. Up to four cards considers every active count; larger machines consider 1, 2, 4, 8, 12, 16 and capacity. Layouts include separate cards, light pairs, stacked pairs, partial shared cores and full stacks. Equal credit weights and extra weight on the better-return row are compared.
Exact moments screen representative layouts. Up to three legal wager levels per layout and automatic horizons of 1, 10, 25, 100, 500 and 1,000 plays are compared. First-visit previews use 64 trajectories. Two leaders per gain/loss pair and, for repeated sessions, one leader per represented layout are checked over the full period on 128 simulations. Up to six finalists per style are independently checked on 1,200 period simulations.
The search chooses gain-goal shares of 5–15% for Reserved, 20–50% for Balanced, 50–100% for Ambitious and 100–300% for Aggressive. These define period comparison thresholds; the gain exit for one visit is scaled to that visit’s available loss allowance. Repeated visits continue according to the schedule even if an earlier balance crosses the period threshold. “Reach the total” means finishing the period at or above it, not touching it at some earlier point.
This is a bounded policy search. “Best tested” means best for the stated objective among the candidates evaluated, not a proven global optimum over every possible card combination, bet or adaptive policy. Reserved ranks an estimated expected turnover using exact RTP. Other rankings use Monte Carlo outcome estimates. Similar candidates can change order under another sample; rare events can be missed.
The schedule: poor, typical and good luck
Each available plan gets a fresh report of 2,000 simulated periods after selection. The chart and schedule show the 10th, 50th and 90th percentiles of the bankroll at each visit. They are realistic poor/typical/good cases, not the worst and best possible results. The curves are daily percentiles and do not form individual player paths. Ties can make fewer than 10% of results lie strictly outside a boundary.
The schedule’s bet, loss allowance and play count summarize a typical visit across all simulations. They are separate medians and should not be treated as a single exact player’s instructions. A second table shows one complete illustrative run, including its actual legal bets, gain exits, loss floors and ending balances. It is the first generated path, without selecting a lucky or unlucky outcome. Actual play instructions recalculate allowances from the current bankroll and remaining sessions.
Island shapes and calculation checks
Separate cards use the original island-stencil library when a shape fits, with compact fallbacks for small or crowded boards. Overlapping cards preserve their shared-number membership when remapped. Historical shape scores are not used to claim better odds: under a fair uniform draw, number identity and shape do not change expected return.
Individual hit probabilities use the hypergeometric distribution. Combined payouts are enumerated exactly when feasible, preserving shared draws and merging identical tickets. Larger portfolios use sampled search previews and fresh shared draws for validation and reports. Pairwise covariance remains exact. Tests check exhaustive small-board results, overlap, payout scaling, island remapping, legal stakes, carry-forward accounting and scheduled loss reserves.
Machine settings and saved setups
Use the machine’s actual paytable and all available denominations. Enter the credits bet displayed with the payouts; the calculator divides by exactly that value. The model assumes linear payout scaling, uniform independent draws and the same normalized table at each denomination. Bonus games, progressives or denomination-specific payouts require their own model. Set minimum/maximum credits and whether every physical card must be active. A style is unavailable if no legal wager fits its budget across the selected visits.
The supplied table is an example. Photo imports need a configured server service and should be checked against the image. Manual calculations run in the browser. Save setup includes the session mode and count; Download includes policies, period estimates, methods and engine version.