Bridge Group Schedule (20 couples, groups of 4)

[Hallsl]

I would like assistance to schedule a church bridge group that has 20 couples that play in groups of 4 couples at a time 8 times during the year.  Each couple hosts once in the first half of the season and once again in the last half of the season.  We would appreciate the schedule to limit the number of times that any couple plays any other couple.  Tough one !!!
Thanks a bunch

[Ian Wakeling]

Jonny,

Just to let you know that I have modified the title to distinguish this thread from another with the same name.

It is possible to have an optimal schedule here, where every couple meets 14 of the other couples exactly once, and the remaining 5 couples exactly twice.   So I think the following schedule should work well for your bridge group.  The 20 couples are labelled A to T.  I have tried to leave as much time as possible between the two occasions when a couple hosts, with a break of at least two rounds between occasions.

  Round   Group 1    Group 2    Group 3    Group 4    Group 5      Hosts
    1    (A C D R)  (E G O Q)  (H I K N)  (B F L M)  (J P S T)  [C G I L T]
    2    (A I M T)  (B E K R)  (D H P Q)  (C F N O)  (G J L S)  [A B D N S]
    3    (D J K T)  (B F O S)  (C E I P)  (A L N Q)  (G H M R)  [K O P Q R]
    4    (E I R S)  (A F G P)  (H O Q T)  (B C J N)  (D K L M)  [E F H J M]

    5    (A K Q S)  (B D E G)  (F H I J)  (C L R T)  (M N O P)  [A G I L N]
    6    (B J M Q)  (C F G K)  (A D I O)  (N P R T)  (E H L S)  [B C O P S]
    7    (A E J N)  (K L O P)  (C H M S)  (D F Q R)  (B G I T)  [E K M R T]
    8    (D G N S)  (E F M T)  (A B H P)  (J K O R)  (C I L Q)  [D F H J Q]

[Hallsl]

Thanks a million times.  You would not believe how much time I spent trying to do this without the benefit of math.  Your quick response was really appreciated.  
Thanks again
Jonny

[Ian Wakeling]

Jonny,

I thought some more about this schedule and noticed that there are 4 couples who at some point return to the same host.  The worst is couple L, who go twice to both Q and S.  That prompted me to look for a better assignment.  So if there is still time, then I suggest you replace the 'Hosts' matrix above with:

[A E N M J]
[T K H F G]
[D S C Q R]
[I P O B L]

[K G J T M]
[Q C D N E]
[A P H R I]
[S F B O L]

For these new host assignments, I have written down the schedule of who goes to which home in which round.  Below an asterisk indicates that a couple hosts.

    1 2 3 4 5 6 7 8
   ----------------
A | * T Q P K D * B
B | M K S * G Q I *
C | A F * B T * H L
D | A H * L G * R S
E | * K C I G * A F
F | M * S P J C R *
G | E * R P * C I S
H | N * R O J E * B
I | N T C * J D * L
J | * G D B * Q A O
K | N * D L * C P O
L | M G Q * T E P *
M | * T R L * Q H F
N | * F Q B M * A S
O | E F S * M D P *
P | J H C * M N * B
Q | E H * O K * R L
R | A K * I T N * O
S | J G * I K E H *
T | J * D O * N I F

This makes it easy to check that all couples host twice and get to go to six different homes.  Hope that helps,

Ian.

[Hallsl]

That's great of you to take a second look at our scheduling issue.  It is not too late to revise our scheduling to your 2nd grouping.
Thanks so much,
Jonny