Shengji is usually played with \[ d = \left\lfloor \frac{p}{2} \right\rfloor \] decks, where $p$ is the number of players in the game. The cards are initially divided into $p$ equally sized groups of size $h$ and one group of size $k$ (the “kitty”). Decks are standard 52-card playing decks, plus a small and a big joker. There are 13 cards in each of the clubs, diamonds, hearts, and spades suits.

For example, with four players, two decks, and an eight-card kitty, each player holds 25 cards.

\[ h = \frac{pd - k}{p} \]

The “landlord” or “leader” gets to choose an $h$-card hand out of $h+k$ cards, so they are more likely to have better hands. But how much more likely?

Card statistics

Trumps

In a standard round, there is a trump suit (i.e. clubs, diamonds, hearts, or spades), as well as a trump number. The total set of trump cards also includes the big and small joker. As a result, there are usually 4 (trump number) + 12 (trump suit, not trump number) + 2 (jokers) = 18 trump cards per deck, vs. 12 cards in each of the non-trump suits.

The following tables list some interesting basic statistics:

Standard trump mechanics

In games where the trump has a trump number and a trump suit:

1 decks, 4 players, 2-card kitty mean min 25% 50% 75% max
number of landlord trump card 5 0 4 5 6 12
number of non-landlord trump card 4.33 0 3 4 5 11
number of non-landlord non-trump card 8.67 2 8 9 10 13
length of non-landlord non-trump suit 2.89 0 2 3 4 9
1 decks, 4 players, 6-card kitty mean min 25% 50% 75% max
number of landlord trump card 5.5 0 4 5 7 13
number of non-landlord trump card 4.17 0 3 4 5 11
number of non-landlord non-trump card 8.33 2 7 8 9 13
length of non-landlord non-trump suit 2.78 0 2 3 4 9
2 decks, 4 players, 4-card kitty mean min 25% 50% 75% max
number of landlord trump card 7 0 5 6 9 20
number of non-landlord trump card 5.67 0 4 5 7 18
number of non-landlord non-trump card 11.33 2 8 9 16 25
length of non-landlord non-trump suit 3.78 0 2 3 5 15
2 decks, 4 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 8 0 5 8 10 21
number of non-landlord trump card 6.33 0 4 6 8 18
number of non-landlord non-trump card 12.67 2 8 12 17 25
length of non-landlord non-trump suit 4.22 0 3 4 6 15
2 decks, 5 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 8.22 0 6 8 10 21
number of non-landlord trump card 6.53 0 5 6 8 18
number of non-landlord non-trump card 13.07 2 10 14 16 25
length of non-landlord non-trump suit 4.35 0 3 4 6 15
3 decks, 6 players, 6-card kitty mean min 25% 50% 75% max
number of landlord trump card 8.57 0 6 9 11 23
number of non-landlord trump card 6.96 0 5 7 9 19
number of non-landlord non-trump card 13.92 2 11 15 17 26
length of non-landlord non-trump suit 4.64 0 3 5 6 17
3 decks, 7 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 8.75 0 7 9 11 23
number of non-landlord trump card 7.03 0 5 7 9 19
number of non-landlord non-trump card 14.06 2 12 15 17 26
length of non-landlord non-trump suit 4.69 0 3 5 6 17
4 decks, 8 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 9.04 0 7 9 11 24
number of non-landlord trump card 7.33 0 6 7 9 19
number of non-landlord non-trump card 14.67 2 13 15 17 26
length of non-landlord non-trump suit 4.89 0 3 5 6 17

No trump mechanics

In games where the trump is declared as “no trump”, so there is a trump number but no trump suit, the landlord advantage in trump cards is substantially reduced.

1 decks, 4 players, 2-card kitty mean min 25% 50% 75% max
number of landlord trump card 1.67 0 1 2 2 6
number of non-landlord trump card 1.44 0 1 1 2 6
number of non-landlord non-trump card 11.56 7 11 12 12 13
length of non-landlord non-trump suit 2.89 0 2 3 4 9
1 decks, 4 players, 6-card kitty mean min 25% 50% 75% max
number of landlord trump card 1.83 0 1 2 3 6
number of non-landlord trump card 1.39 0 1 1 2 6
number of non-landlord non-trump card 11.11 6 10 11 12 13
length of non-landlord non-trump suit 2.78 0 2 3 4 9
2 decks, 4 players, 4-card kitty mean min 25% 50% 75% max
number of landlord trump card 2.33 0 1 2 3 11
number of non-landlord trump card 1.89 0 1 2 3 9
number of non-landlord non-trump card 15.11 6 11 12 22 26
length of non-landlord non-trump suit 3.78 0 2 3 5 15
2 decks, 5 players, 3-card kitty mean min 25% 50% 75% max
number of landlord trump card 2.67 0 2 3 4 11
number of non-landlord trump card 2.17 0 1 2 3 10
number of non-landlord non-trump card 17.33 6 12 19 22 26
length of non-landlord non-trump suit 4.33 0 3 4 6 15
2 decks, 4 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 2.67 0 2 2 4 11
number of non-landlord trump card 2.11 0 1 2 3 10
number of non-landlord non-trump card 16.89 6 11 15 23 26
length of non-landlord non-trump suit 4.22 0 3 4 6 15
2 decks, 5 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 2.74 0 2 3 4 11
number of non-landlord trump card 2.18 0 1 2 3 10
number of non-landlord non-trump card 17.42 6 12 18 21 26
length of non-landlord non-trump suit 4.35 0 3 4 6 15
3 decks, 6 players, 6-card kitty mean min 25% 50% 75% max
number of landlord trump card 2.86 0 2 3 4 11
number of non-landlord trump card 2.32 0 1 2 3 10
number of non-landlord non-trump card 18.56 6 16 19 23 26
length of non-landlord non-trump suit 4.64 0 3 5 6 17
3 decks, 7 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 2.92 0 2 3 4 12
number of non-landlord trump card 2.34 0 1 2 3 10
number of non-landlord non-trump card 18.75 6 17 19 22 26
length of non-landlord non-trump suit 4.69 0 3 5 6 17
4 decks, 8 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump card 3.01 0 2 3 4 12
number of non-landlord trump card 2.44 0 1 2 3 11
number of non-landlord non-trump card 19.56 6 18 20 23 26
length of non-landlord non-trump suit 4.89 0 3 5 6 17

Valid trick formats

2 decks, 4 players, 4-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 0.45 0 0 0 1 7
number of landlord non-trump pairs 0.9 0 0 0 2 9
number of non-landlord trump pairs in one hand 0.34 0 0 0 0 6
number of non-landlord non-trump pairs in one hand 0.67 0 0 0 1 8
number of non-landlord trump pairs in all hands 1.01 0 0 0 2 11
number of non-landlord non-trump pairs in all hands 2.02 0 0 0 5 16
2 decks, 4 players, 4-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.03 0 0 0 0 4
number of landlord non-trump 2x2 tractors 0.06 0 0 0 0 4
number of non-landlord trump 2x2 tractors in one hand 0.02 0 0 0 0 3
number of non-landlord non-trump 2x2 tractors in one hand 0.03 0 0 0 0 3
number of non-landlord trump 2x2 tractors in all hands 0.05 0 0 0 0 5
number of non-landlord non-trump 2x2 tractors in all hands 0.09 0 0 0 0 4
number of landlord trump 2x3 tractors 0 0 0 0 0 3
number of landlord non-trump 2x3 tractors 0 0 0 0 0 2
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 2
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 2
number of non-landlord trump 2x3 tractors in all hands 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in all hands 0 0 0 0 0 3
2 decks, 4 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 0.75 0 0 0 1 7
number of landlord non-trump pairs 1.5 0 0 0 3 10
number of non-landlord trump pairs in one hand 0.49 0 0 0 1 6
number of non-landlord non-trump pairs in one hand 0.97 0 0 0 2 8
number of non-landlord trump pairs in all hands 1.46 0 0 0 3 11
number of non-landlord non-trump pairs in all hands 2.92 0 0 0 6 16
2 decks, 4 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.06 0 0 0 0 4
number of landlord non-trump 2x2 tractors 0.11 0 0 0 0 6
number of non-landlord trump 2x2 tractors in one hand 0.02 0 0 0 0 4
number of non-landlord non-trump 2x2 tractors in one hand 0.04 0 0 0 0 3
number of non-landlord trump 2x2 tractors in all hands 0.07 0 0 0 0 5
number of non-landlord non-trump 2x2 tractors in all hands 0.13 0 0 0 0 5
number of landlord trump 2x3 tractors 0 0 0 0 0 3
number of landlord non-trump 2x3 tractors 0.01 0 0 0 0 5
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 2
number of non-landlord trump 2x3 tractors in all hands 0 0 0 0 0 4
number of non-landlord non-trump 2x3 tractors in all hands 0 0 0 0 0 3
2 decks, 5 players, 3-card kitty mean min 25% 50% 75% max
number of landlord trump card 8 0 6 8 10 21
number of non-landlord trump card 6.5 0 4 6 8 18
number of non-landlord non-trump card 13 2 9 14 16 25
length of non-landlord non-trump suit 4.33 0 3 4 6 15
2 decks, 5 players, 3-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 0.77 0 0 0 1 7
number of landlord non-trump pairs 1.54 0 0 1 3 10
number of non-landlord trump pairs in one hand 0.52 0 0 0 1 6
number of non-landlord non-trump pairs in one hand 1.04 0 0 1 2 8
number of non-landlord trump pairs in all hands 1.69 0 0 1 3 11
number of non-landlord non-trump pairs in all hands 3.38 0 0 4 6 16
2 decks, 5 players, 3-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.06 0 0 0 0 4
number of landlord non-trump 2x2 tractors 0.1 0 0 0 0 6
number of non-landlord trump 2x2 tractors in one hand 0.02 0 0 0 0 4
number of non-landlord non-trump 2x2 tractors in one hand 0.04 0 0 0 0 3
number of non-landlord trump 2x2 tractors in all hands 0.07 0 0 0 0 5
number of non-landlord non-trump 2x2 tractors in all hands 0.13 0 0 0 0 5
number of landlord trump 2x3 tractors 0 0 0 0 0 3
number of landlord non-trump 2x3 tractors 0.01 0 0 0 0 5
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 2
number of non-landlord trump 2x3 tractors in all hands 0 0 0 0 0 4
number of non-landlord non-trump 2x3 tractors in all hands 0 0 0 0 0 3
2 decks, 5 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 0.84 0 0 0 1 7
number of landlord non-trump pairs 1.68 0 0 2 3 10
number of non-landlord trump pairs in one hand 0.53 0 0 0 1 6
number of non-landlord non-trump pairs in one hand 1.06 0 0 1 2 8
number of non-landlord trump pairs in all hands 1.8 0 0 2 3 11
number of non-landlord non-trump pairs in all hands 3.61 0 0 4 6 16
2 decks, 5 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.06 0 0 0 0 4
number of landlord non-trump 2x2 tractors 0.1 0 0 0 0 6
number of non-landlord trump 2x2 tractors in one hand 0.02 0 0 0 0 4
number of non-landlord non-trump 2x2 tractors in one hand 0.04 0 0 0 0 3
number of non-landlord trump 2x2 tractors in all hands 0.07 0 0 0 0 5
number of non-landlord non-trump 2x2 tractors in all hands 0.13 0 0 0 0 5
number of landlord trump 2x3 tractors 0 0 0 0 0 3
number of landlord non-trump 2x3 tractors 0.01 0 0 0 0 5
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 2
number of non-landlord trump 2x3 tractors in all hands 0 0 0 0 0 4
number of non-landlord non-trump 2x3 tractors in all hands 0 0 0 0 0 3
3 decks, 6 players, 6-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 0.98 0 0 1 2 7
number of landlord non-trump pairs 1.95 0 0 2 3 10
number of non-landlord trump pairs in one hand 0.63 0 0 0 1 6
number of non-landlord non-trump pairs in one hand 1.26 0 0 1 2 9
number of non-landlord trump pairs in all hands 2.41 0 0 2 4 15
number of non-landlord non-trump pairs in all hands 4.82 0 0 5 7 26
number of landlord trump triples 0.02 0 0 0 0 3
number of landlord non-trump triples 0.04 0 0 0 0 3
number of non-landlord trump triples in one hand 0.01 0 0 0 0 3
number of non-landlord non-trump triples in one hand 0.02 0 0 0 0 3
number of non-landlord trump triples in all hands 0.05 0 0 0 0 5
number of non-landlord non-trump triples in all hands 0.1 0 0 0 0 7
3 decks, 6 players, 6-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.07 0 0 0 0 5
number of landlord non-trump 2x2 tractors 0.13 0 0 0 0 6
number of non-landlord trump 2x2 tractors in one hand 0.03 0 0 0 0 4
number of non-landlord non-trump 2x2 tractors in one hand 0.05 0 0 0 0 4
number of non-landlord trump 2x2 tractors in all hands 0.12 0 0 0 0 5
number of non-landlord non-trump 2x2 tractors in all hands 0.2 0 0 0 0 6
number of landlord trump 2x3 tractors 0.01 0 0 0 0 4
number of landlord non-trump 2x3 tractors 0.01 0 0 0 0 5
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord trump 2x3 tractors in all hands 0.01 0 0 0 0 4
number of non-landlord non-trump 2x3 tractors in all hands 0.01 0 0 0 0 3
number of landlord trump 3x2 tractors 0 0 0 0 0 1
number of landlord non-trump 3x2 tractors 0 0 0 0 0 2
number of non-landlord trump 3x2 tractors in one hand 0 0 0 0 0 1
number of non-landlord non-trump 3x2 tractors in one hand 0 0 0 0 0 1
number of non-landlord trump 3x2 tractors in all hands 0 0 0 0 0 1
number of non-landlord non-trump 3x2 tractors in all hands 0 0 0 0 0 1
number of landlord trump 3x3 tractors 0 0 0 0 0 0
number of landlord non-trump 3x3 tractors 0 0 0 0 0 1
number of non-landlord trump 3x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord non-trump 3x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord trump 3x3 tractors in all hands 0 0 0 0 0 0
number of non-landlord non-trump 3x3 tractors in all hands 0 0 0 0 0 0
3 decks, 7 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 1.05 0 0 1 2 7
number of landlord non-trump pairs 2.11 0 0 2 3 10
number of non-landlord trump pairs in one hand 0.66 0 0 0 1 6
number of non-landlord non-trump pairs in one hand 1.32 0 0 1 2 9
number of non-landlord trump pairs in all hands 2.77 0 0 3 4 15
number of non-landlord non-trump pairs in all hands 5.54 0 0 5 8 26
number of landlord trump triples 0.03 0 0 0 0 3
number of landlord non-trump triples 0.06 0 0 0 0 3
number of non-landlord trump triples in one hand 0.01 0 0 0 0 3
number of non-landlord non-trump triples in one hand 0.03 0 0 0 0 3
number of non-landlord trump triples in all hands 0.07 0 0 0 0 5
number of non-landlord non-trump triples in all hands 0.14 0 0 0 0 7
3 decks, 7 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.08 0 0 0 0 5
number of landlord non-trump 2x2 tractors 0.14 0 0 0 0 6
number of non-landlord trump 2x2 tractors in one hand 0.03 0 0 0 0 4
number of non-landlord non-trump 2x2 tractors in one hand 0.05 0 0 0 0 4
number of non-landlord trump 2x2 tractors in all hands 0.13 0 0 0 0 5
number of non-landlord non-trump 2x2 tractors in all hands 0.23 0 0 0 0 6
number of landlord trump 2x3 tractors 0.01 0 0 0 0 4
number of landlord non-trump 2x3 tractors 0.01 0 0 0 0 5
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord trump 2x3 tractors in all hands 0.01 0 0 0 0 4
number of non-landlord non-trump 2x3 tractors in all hands 0.01 0 0 0 0 3
number of landlord trump 3x2 tractors 0 0 0 0 0 1
number of landlord non-trump 3x2 tractors 0 0 0 0 0 2
number of non-landlord trump 3x2 tractors in one hand 0 0 0 0 0 1
number of non-landlord non-trump 3x2 tractors in one hand 0 0 0 0 0 1
number of non-landlord trump 3x2 tractors in all hands 0 0 0 0 0 1
number of non-landlord non-trump 3x2 tractors in all hands 0 0 0 0 0 1
number of landlord trump 3x3 tractors 0 0 0 0 0 0
number of landlord non-trump 3x3 tractors 0 0 0 0 0 1
number of non-landlord trump 3x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord non-trump 3x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord trump 3x3 tractors in all hands 0 0 0 0 0 0
number of non-landlord non-trump 3x3 tractors in all hands 0 0 0 0 0 0
4 decks, 8 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump pairs 1.17 0 0 1 2 9
number of landlord non-trump pairs 2.34 0 0 2 4 11
number of non-landlord trump pairs in one hand 0.73 0 0 0 1 7
number of non-landlord non-trump pairs in one hand 1.46 0 0 1 2 9
number of non-landlord trump pairs in all hands 3.47 0 1 3 5 19
number of non-landlord non-trump pairs in all hands 6.94 0 2 6 11 35
number of landlord trump triples 0.05 0 0 0 0 4
number of landlord non-trump triples 0.1 0 0 0 0 5
number of non-landlord trump triples in one hand 0.02 0 0 0 0 3
number of non-landlord non-trump triples in one hand 0.05 0 0 0 0 4
number of non-landlord trump triples in all hands 0.14 0 0 0 0 6
number of non-landlord non-trump triples in all hands 0.29 0 0 0 0 9
number of landlord trump 4-tuples 0 0 0 0 0 2
number of landlord non-trump 4-tuples 0 0 0 0 0 2
number of non-landlord trump 4-tuples in one hand 0 0 0 0 0 1
number of non-landlord non-trump 4-tuples in one hand 0 0 0 0 0 1
number of non-landlord trump 4-tuples in all hands 0 0 0 0 0 2
number of non-landlord non-trump 4-tuples in all hands 0 0 0 0 0 3
4 decks, 8 players, 8-card kitty mean min 25% 50% 75% max
number of landlord trump 2x2 tractors 0.1 0 0 0 0 5
number of landlord non-trump 2x2 tractors 0.17 0 0 0 0 6
number of non-landlord trump 2x2 tractors in one hand 0.04 0 0 0 0 4
number of non-landlord non-trump 2x2 tractors in one hand 0.07 0 0 0 0 4
number of non-landlord trump 2x2 tractors in all hands 0.18 0 0 0 0 6
number of non-landlord non-trump 2x2 tractors in all hands 0.32 0 0 0 0 7
number of landlord trump 2x3 tractors 0.01 0 0 0 0 4
number of landlord non-trump 2x3 tractors 0.01 0 0 0 0 5
number of non-landlord trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord non-trump 2x3 tractors in one hand 0 0 0 0 0 3
number of non-landlord trump 2x3 tractors in all hands 0.01 0 0 0 0 4
number of non-landlord non-trump 2x3 tractors in all hands 0.01 0 0 0 0 4
number of landlord trump 3x2 tractors 0 0 0 0 0 2
number of landlord non-trump 3x2 tractors 0 0 0 0 0 2
number of non-landlord trump 3x2 tractors in one hand 0 0 0 0 0 1
number of non-landlord non-trump 3x2 tractors in one hand 0 0 0 0 0 1
number of non-landlord trump 3x2 tractors in all hands 0 0 0 0 0 2
number of non-landlord non-trump 3x2 tractors in all hands 0 0 0 0 0 2
number of landlord trump 3x3 tractors 0 0 0 0 0 1
number of landlord non-trump 3x3 tractors 0 0 0 0 0 1
number of non-landlord trump 3x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord non-trump 3x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord trump 3x3 tractors in all hands 0 0 0 0 0 0
number of non-landlord non-trump 3x3 tractors in all hands 0 0 0 0 0 1
number of landlord trump 4x2 tractors 0 0 0 0 0 1
number of landlord non-trump 4x2 tractors 0 0 0 0 0 1
number of non-landlord trump 4x2 tractors in one hand 0 0 0 0 0 0
number of non-landlord non-trump 4x2 tractors in one hand 0 0 0 0 0 0
number of non-landlord trump 4x2 tractors in all hands 0 0 0 0 0 0
number of non-landlord non-trump 4x2 tractors in all hands 0 0 0 0 0 0
number of landlord trump 4x3 tractors 0 0 0 0 0 0
number of landlord non-trump 4x3 tractors 0 0 0 0 0 0
number of non-landlord trump 4x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord non-trump 4x3 tractors in one hand 0 0 0 0 0 0
number of non-landlord trump 4x3 tractors in all hands 0 0 0 0 0 0
number of non-landlord non-trump 4x3 tractors in all hands 0 0 0 0 0 0