rigorous seating

Seating concerns for rigorous players of the game Seven for Seven.

The sequence of players as they are seated around the table greatly influences who might be a partner of whom; whether partners will be in the trio or in the quartet; and how often these might occur. Rigorous players will take an interest in this matter.

This page starts by looking at three sets of seven hands each. Among the hands of each set, the player sequence is constant, but from one set to the next the sequence differs. A hypothetical situation is assumed: within each set, each player will be the captain once. In reality, this is not likely to happen, because any player is eligible to be the captain in any hand, and in a realistic set there will often be a player who is captain more than once, and another player who is never captain. However, the assumption does provide a balanced basis for examination.

Within each set entabulated below, players are seated clockwise around the table:

Set 1: Amber, Betty, Cindy, Diana, Ellie, Flora, Greta. (as on the home page)
Set 2: Amber, Cindy, Ellie, Greta, Betty, Diana, Flora.
Set 3: Amber, Diana, Greta, Cindy, Flora, Betty, Ellie.

In the tables, names are often abbreviated to their initials.

set 1
trio		quartet
captain	lieutenants	resisters
A	C, F	B, D, E, G
B	D, G	C, E, F, A
C	E, A	D, F, G, B
D	F, B	E, G, A, C
E	G, C	F, A, B, D
F	A, D	G, B, C, E
G	B, E	A, C, D, F

set 2
trio		quartet
captain	lieutenants	resisters
A	E, D	C, G, B, F
C	G, F	E, B, D, A
E	B, A	G, D, F, C
G	D, C	B, F, A, E
B	F, E	D, A, C, G
D	A, G	F, C, E, B
F	C, B	A, E, G, D

set 3
trio		quartet
captain	lieutenants	resisters
A	G, B	D, C, F, E
D	C, E	G, F, B, A
G	F, A	C, B, E, D
C	B, D	F, E, A, G
F	E, G	B, A, D, C
B	A, C	E, D, G, F
E	D, F	A, G, C, B

The table below summarizes quantitative information from the tables above; here is an example of how to interpret it. With the assumption that each player is captain once in each set, then in set 1:

Betty is never Amber's trio partner, but is her quartet partner once, and is her opponent six times.
Cindy is Amber's trio partner twice, her quartet parter three times, and her opponent twice.
Diana is Amber's trio partner once, her quartet partner twice, and hter opponent four times.

The rest of the table is read similarly.

Within any one set, the distribution of partnerships is quite uneven. However, if all three sets (a total of 21 hands) are played, the totals work out evenly for all of the six other players.

Amber and the other players
number of times each player is: T = Amber's trio partner; Q = Amber's quartet partner; x = Amber's opponent
	Betty			Cindy			Diana			Ellie			Flora			Greta
	T	Q	x	T	Q	x	T	Q	x	T	Q	x	T	Q	x	T	Q	x
set 1	0	1	6	2	3	2	1	2	4	1	2	4	2	3	2	0	1	6
set 2	1	2	4	0	1	6	2	3	2	2	3	2	0	1	6	1	2	4
set 3	2	3	2	1	2	4	0	1	6	0	1	6	1	2	4	2	3	2
totals	3	6	12	3	6	12	3	6	12	3	6	12	3	6	12	3	6	12

Because of the symmetry of the arrangement, corresponding charts for other players will be analogous. A further aspect of the symmetry is reflectivity: Betty and Greta have the same numbers; as do Cindy and Flora; Diana and Ellie.

There is a further point for rigorous players. Recall from above:

Set 1: Amber, Betty, Cindy, Diana, Ellie, Flora, Greta
Set 2: Amber, Cindy, Ellie, Greta, Betty, Diana, Flora
Set 3: Amber, Diana, Greta, Cindy, Flora, Betty, Ellie

Only three of the six players (Greta, Flora, Ellie) ever appear as Amber's immediate left-hand opponents. To achieve balance, three additional seating arrangements can be played, for a lengthy session of 42 hands:

Set 4: Amber, Ellie, Betty, Flora, Cindy, Greta, Diana (reverse of set 3)
Set 5: Amber, Flora, Diana, Betty, Greta, Ellie, Cindy (reverse of set 2)
Set 6: Amber, Greta, Flora, Ellie, Diana, Cindy, Betty (reverse of set 1)

Now Amber's immediate left-hand opponents include Diana, Cindy, and Betty. Other players are similarly balanced.

Comment. Because there are 7 hands in a set of 747, perhaps a set ought to be called a 74747.