# Aotenjou

**Aotenjou** 「青天井」 is a rarely used scoring system where all limits are removed.

Per the scoring equation, han is factored exponentially in the equation:

**Basic points = fu × 2**^{(2 + han)}

Normally, the point equation has a cap applied at mangan (2000 base points), with higher hands being multiples of mangan.

In aotenjou, this cap is removed - each han will double your score indefinitely. Due to the exponential nature of the scoring system, this can produce astronomically large point values. The modifiers for tsumo, ron, and dealer seat are then applied as normal, increasing the score further.

In addition, yakuman are treated as 13 han yaku, so they can stack with regular yaku and dora. As with regular yaku, any yaku that are implied by the completion of a yakuman are not scored. For example, a suuankou hand does not score sanankou or toitoi, because these are prerequisites for suuankou.

The points can get very large quickly. An 8 han 30 fu ron, worth 16,000 normally, is worth 122,880 in aotenjou. A yakuman ron (40 fu) is 32,000 normally and 5,242,880 in aotenjou. For this very reason, the limits of mangan and yakuman are applied in standard rules; the numbers are simply too large for practical use.

## Extreme scoring examples

- +
- This hand is scored as 13 han 40 fu (ron) or 14 han 30 fu (tsumo, counting mentsumo). By dealer, it is worth 7,864,400 (ron) or 1,966,080 all (tsumo).

- Agari: or
- With aotenjou, every aspect of this hand is considered, not just the yakuman patterns. This hand also includes the regular yaku of toitoi and sanankou as well as two or three yakuhai when applicable. This hand stands at 31 han and 50 fu via tsumo:

Any dora to these examples definitely increases the hand even further.

### Upper limit

Because a finite-sized hand can only score so many han and fu, there is a maximum score that a single hand can obtain in aotenjou.

In a hypothetical ruleset where double yakuman exist and are scored as 26 han, the maximum scoring hand would be as follows:

Assume the player is dealer, it's the East round, and has the yaku of riichi, ippatsu, tsumo, haitei.

This hand's fu value would be calculated as:

- base fu: 20 fu
- pair wait: 2 fu
- tsumo: 2 fu
- composition of hand:

Total 154 fu, round up to 160

The han value of this hand would be calculated as:

- tsuuiisou: 13 han
- suukantsu: 13 han
- suuankou tanki: 26 han
- riichi: 1 han
- ippatsu: 1 han
- tsumo: 1 han
- haitei raoyue: 1 han
- double east: 2 han
- dragons: 2 han
- shousangen: 2 han
- 40 dora: 40 han

Total 102 han

The base points value of this hand would be 3.25 × 10^{33} and the dealer would receive 6.50 × 10^{33} points from each player, for a total of 1.95 × 10^{34} points.

The highest possible fu count is 170, as in the following example for the south player in the south round:

- base fu: 20
- closed ron: 10
- pair wait: 2
- composition of hand:

Total 164, round up to 170

## Yakuman/Yaku Compability

Certain yakuman imply certain yaku, and thus they cannot be scored together:

- Suuankou: Sanankou, toitoi
- Kokushi: Honroutou
- Daisangen: The three dragon yakuhai
- Tsuuiisou: Honroutou, chanta
- Daisuushii: Shousuushii, round wind, seat wind, toitoi
- Chinroutou: Honroutou, junchan, chanta, toitoi
- Ryuuiisou: Honitsu, If hatsu is forbidden: chinitsu
- Chuuren poutou: Chinitsu, honitsu
- Tenhou and chiihou: Mentsumo
- Suukantsu: Sankantsu, toitoi

These may be subject to further rule variation. For example, if daichiishin (Tsuuiisou + chiitoitsu) is considered an entirely separate yakuman, then regular tsuuiisou cannot score toitoi. The same applies to suuankou + mentsumo if suuankou tanki is its own yakuman.

## External links

**青天井**in Japanese Wikipedia