A return you can compute
Hanabi is a 5×3, ten-line festival slot, and unlike most slots it has no reel strips. Each of the fifteen cells is an independent weighted draw from the committed seed pair. Designing Hanabi laid out the weights and the free-spin ladder and then drew a line: the trigger rate could be checked by hand, but the return was a simulation result the Journal could only quote. That line was drawn in the wrong place. Independent cells make the return a sum of small exact pieces, and this article adds them up.
Three facts make an exact answer possible. The expected value of a spin is the sum of the expected values of its ten lines and its scatter pay, whatever the overlaps between lines. Every line sees five independent cells with the same weights, so one line’s expected pay is a sum over 9⁵ = 59,049 weighted symbol sequences. And the scatter count is binomial. The free spins then multiply the same per-spin value by a ladder whose length is a small Markov chain. None of it needs a single simulated spin.
Where the 95.81% comes from
| SYMBOL | WEIGHT | PAYS 3 / 4 / 5 | A GIVEN LINE PAYS IT | SHARE OF RETURN |
|---|---|---|---|---|
| 金魚 Kingyo | 4 | ×3 / ×10 / ×40 | 0.141% | 5.99% |
| 提灯 Chochin | 5 | ×2 / ×6 / ×25 | 0.215% | 6.19% |
| 団扇 Uchiwa | 6 | ×1.5 / ×4 / ×15 | 0.311% | 6.59% |
| 氷 Kakigori | 8 | ×1 / ×3 / ×10 | 0.576% | 9.20% |
| 風鈴 Furin | 9 | ×0.8 / ×2.5 / ×8 | 0.751% | 10.17% |
| 麺 Yakisoba | 12 | ×0.6 / ×1.6 / ×5 | 1.478% | 15.57% |
| 鈴 Suzu | 12 | ×0.5 / ×1.2 / ×4 | 1.478% | 12.46% |
| All line wins | 4.948% | 66.17% | ||
| 祭 scatter pays | 2 | ×2 / ×5 / ×20 | 1.193% of spins | 2.87% |
| Free-spin feature | 10+ spins, ×1 → ×5 | 1 spin in 83.8 | 26.77% | |
| Total | 95.81% |
Pays are multiples of the total bet. “A given line pays it” is the chance that one specific line wins with that symbol on a spin; ten lines are evaluated per spin. Share of return = ten lines × the line’s expected pay, as a percentage of the bet. Kingyo includes the pure-wild runs that pay as 金魚.
The table has one surprise. The biggest contributors are the two cheapest symbols, 麺 and 鈴, which pay at most ×5 and ×4 for five of a kind yet supply more than a quarter of the return between them. The ×40 金魚 line supplies 6%. Weight, not the printed top prize, decides where a slot’s return lives — the same lesson Tairyō’s paytable teaches from the other side.
The free-spin ladder, priced
Three or more scatters among fifteen cells, each a scatter with chance 2/61, is a binomial event: 1.075% for exactly three, 0.109% for four and 0.0086% for five or more, 1.193% in all, or one base spin in 83.8. The feature that follows is worth the per-spin value of the base game, 0.690 of the bet before any multiplier, times the sum of the ladder’s multipliers over the spins actually played.
| FEATURE LENGTH | CHANCE | LADDER SUM |
|---|---|---|
| 10 spins, no retrigger | 88.69% | 30 |
| 14 spins (one retrigger) | 10.21% | 50 |
| 18 spins (two) | 0.998% | 70 |
| 22 spins (the cap) | 0.104% | 90 |
Each feature spin retriggers with the same 1.193%. Ten spins carry multipliers ×1, ×1, ×2, ×2, ×3, ×3, ×3, ×5, ×5, ×5 (sum 30); every later spin is ×5. Average length 10.50 spins, average ladder sum 32.50, so the feature is worth 0.690 × 32.50 = ×22.44 the bet on average.
Most of the feature’s value sits in the last three spins of the base ten, which carry ×5 each: fifteen of the thirty multiplier points. That is the design the name promises — the firework builds to the finale — and it also means the feature’s outcome is uneven. In two million replayed features the middle one paid ×16.8, 29.0% paid under ×10, 12.5% under ×5, 7.6% between ×50 and ×100 and 0.98% ×100 or more. The largest paid ×513.8. None came near the ×2,000 cap, so the cap does not move the return by any amount the replay could see.
What a spin and a session look like
| BASE SPIN PAYS | SHARE OF SPINS |
|---|---|
| Nothing | 66.57% |
| Something, but less than the bet | 11.72% |
| ×1 to under ×5 | 18.72% |
| ×5 to under ×20 | 2.90% |
| ×20 or more | 0.098% |
Replay of the engine’s evaluation over 20,000,000 base spins drawn from the published weights; its mean, 0.6906 of the bet, matches the exact 0.6904. The standard deviation of one full round, feature included, is 3.9 bets.
The second row deserves its own line. One spin in nine lights up, plays its win animation and pays back less than it cost — a loss the screen presents as a win. Slot researchers call these losses disguised as wins, and the machine zone describes what they do to a player’s sense of how a session is going. Counting only spins that return at least the stake, the hit rate is 21.7%, not 33.4%.
At a 4.19% expected loss per spin, 100 spins cost 4.2 bets on average. The average hides the shape: the feature’s rare large payouts pull the mean up, so the median 100-spin session in the replay ended 12.9 bets down and 36% of sessions finished ahead. Over 1,000 spins the expected loss is 41.9 bets, the median session is 52.7 down, and 33% still finish ahead. Risk of ruin turns that spread into how long a bankroll lasts.
Checking a round
- From the round’s fairness panel take the revealed server seed, the client seed and the nonce.
- For cursors 0 to 14 compute HMAC-SHA256 keyed by the server seed over “clientSeed-nonce-cursor”, take the first eight bytes as a fraction of 2⁶⁴, multiply by 61 and walk the weight table (3 wild, 2 scatter, 4, 5, 6, 8, 9, 12, 12) to get each cell; cell = reel × 3 + row.
- Evaluate the ten lines and the scatters as above. If the feature triggered, free spin j uses cursors 15 + 15j to 29 + 15j, with the ladder multiplier for that spin.
The three steps fit in a short script, and one rounding, not many explains why the fraction must be taken exactly as the server takes it. Checking the grid proves the round was fixed before you pressed Spin. The 95.81% is a different kind of claim — a property of the weights, not of any one round — and this article is the proof of it.
What is the RTP of the Hanabi slot?
Exactly 95.81%, computed from the symbol weights: 66.17% from line wins, 2.87% from scatter pays and 26.77% from the free-spin feature. The engine header’s 96.1% was a simulation estimate about three-tenths of a point too high.
How often do Hanabi free spins trigger?
On 1.193% of base spins, one in 83.8: three or more 祭 scatters among fifteen cells, each a scatter with chance 2/61.
How much are Hanabi free spins worth?
×22.44 the bet on average. The middle feature pays ×16.8; 29% pay under ×10 and about 1% pay ×100 or more. The feature runs 10 spins, 14 with one retrigger (10.2% of features) and rarely more.
Which Hanabi symbol pays the most over time?
麺 Yakisoba, at 15.57% of the return, followed by 鈴 Suzu at 12.46%. The two cheapest symbols land most often; the ×40 金魚 contributes 5.99%.
How often does a spin pay?
About 33.4% of base spins pay something, but 11.7% pay less than the bet, so 21.7% return at least the stake.
Can a Hanabi round hit the ×2,000 cap?
In principle; in two million replayed features the largest paid ×513.8, so the cap has no measurable effect on the return.
- Betkyo engine source: hanabi/engine.ts — HANABI_SYMBOLS, HANABI_LINES, evalHanabiSpin(), HANABI_SCATTER_PAY100, HANABI_FS_START / RETRIG / CAP, hanabiFsMult(), HANABI_WIN_CAP100, playHanabiRound(); house port HanabiEngine.kt with the same tables and settlement at totalMult100 ÷ 100
- Exact calculation written for this article: one line’s expected pay summed over all 9⁵ weighted symbol sequences under the engine’s line rule; the binomial scatter count over 15 cells at 2/61; the free-spin length as a Markov chain over the retrigger rule; feature value = per-spin value × expected ladder sum
- Replay written for this article: 20,000,000 base spins and 2,000,000 features drawn from the published weights and evaluated by the engine’s rules (mean 0.6906 vs exact 0.6904 per spin; feature ×22.439 vs exact ×22.440), for the hit rate, the pay spread, the feature distribution and the session outcomes
