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