Math problem

[John Pratt]

Math has never been my strong suit and most assuredly never will be, however; maybe one of you can help me with this problem. I have eight people that need to be broken down into two groups of four. These eight people need to be broken into these groups eight separate times. Is there a math formula or a system to ensure that each person is placed into a group (over the course of the eight 4-man groups) so that they have an opportunity to be placed with the other seven people an equal number of times?

For you math guys this is probably simple, but I just got a headache reading it back to myself.

John

[Sean Hughto]

Give them all a number. Then:

1234 5678
2345 6781
3456 7812
4567 8123
5678 1234
6781 2345
7812 3456
8123 4567

They'll each number will see the others 6 times.

I'm no math whiz, so I could be wrong, but it's the best I could do.

[John Coloccia]

Sean, in your scheme you will never be paired with the number that is 4 away from your. For example, 1 never sees 5.

[Sean Hughto]

d'oh............!

[John Coloccia]

I'm not convinced it's possible, actually. I suspect that brushing off my old graph theory books might be of some benefit here :)

[Eric DeSilva]

The first guy has to be in eight groups, right? That means he has 3 x 8, or 24 possible "slots" to fill in his groups and divide evenly among the 7 other people. But 7 doesn't go into 24 evenly. So I don't think it works.

[Steve Meliza]

1234 5678
8123 4567
7812 3456
6781 2345
8234 1567
7823 4156
6782 3415
5678 2341 (repeat)

I think this works. Note the repeat grouping, that's because you only need 7 sets of groupings (I think).

[John Coloccia]

1234 5678
8123 4567
7812 3456
6781 2345
8234 1567
7823 4156
6782 3415
5678 2341

As a quick look, 1 is paired with 2,3,4 four times, but 5 only 3 times. I suspect the same kind of pathology affects other numbers. I really don't think it's possible, but now I've gone down the rabbit hole trying to figure out the general case and all of the various rules to develop an intuition about this kind of problem.....way way down the rabbit hole. I knew I should have skipped this thread. LOL

[Steve Friedman]

What you are describing is identical to a tennis doubles round robin tournament. I have run a number (for charities) and there are very simple downloadable programs that do this for you. The last one I used was called "Gameplan" by Greencourt Software. It was $25, but I would imagine similar programs are available as free downloads these days. I would run it for you if I still had the program.

The programs try to optimize how many times each player plays with every other player, how many times each player plays against every other player, and avoids duplicate matches. It's harder than you might imagine to make things come out even.

Steve

[Myk Rian]

We used this web site to create our softball schedule.
Something here might work.
http://quickleague.org/Home.aspx

[John Pratt]

Now you can possibly see where I am having troubles. It is not so important that the people are grouped exactly the same number of times with different people in groups or that each person sees every other person the exact number of times but it needs to be close (ie. 4-5 times each, but not 2 for one person and sees another 6 times). If we changed the number of two four-person goups to 7 instead of 8 iterations of two four-person groups, would that make it easier or more difficult?

[Steve Friedman]

This is addictive, but here it is - 7 rounds. Everyone is paired with everyone else exactly 3 times (I think). I would probably mix up the rounds so that (for example) 1 & 2 are not together 3 times in a row). Doesn't work with 8 rounds - only works in multiples of 7.

Round 1: 1234 5678
Round 2: 1258 3467
Round 3: 1267 3458
Round 4: 1357 2468
Round 5: 1368 2457
Round 6: 1456 2378
Round 7: 1478 2356

Now I can go to sleep! I love puzzles.

Steve

[John Pratt]

Thanks Steve, That does seem to work. I appreciate the brain power (mine was about to explode everytime I tried figuring this out).

[Steve Baumgartner]

What you are describing is identical to a tennis doubles round robin tournament. I have run a number (for charities) and there are very simple downloadable programs that do this for you. The last one I used was called "Gameplan" by Greencourt Software. It was $25, but I would imagine similar programs are available as free downloads these days. I would run it for you if I still had the program.

The programs try to optimize how many times each player plays with every other player, how many times each player plays against every other player, and avoids duplicate matches. It's harder than you might imagine to make things come out even.

Steve

Agreed! I used to set up round robin rotations for our tennis club and found that there are only certain combinations of number of players and number of rounds for which it is even theoretically possible to create a perfect rotation. I wrote a program that brute-force tried all the possible permutations (yes, it took a long time to run!) and it confirmed that not all combinations have any possible solutions!

[John Pratt]

I like the idea of using the same format of tennis double in the future. However, doesn't the fact that there are four people grouped together instead of two, throw of the number combinations since two people are not paired together and you want each person to play an equal number of times with the other seven people. I am not sure how Steve did it other than trial and error.

[Steve Friedman]

I like the idea of using the same format of tennis double in the future. However, doesn't the fact that there are four people grouped together instead of two, throw of the number combinations since two people are not paired together and you want each person to play an equal number of times with the other seven people. I am not sure how Steve did it other than trial and error.
I did this based on a combination of trying to remember how the computer program did it plus trial and error. It would take forever to do that by hand for tennis doubles.

I no longer have the program that did the tennis doubles grid, but do have about 30 mixed doubles grids (which can be modified for regular doubles) saved and am happy to share. I have them for 16 to 32 players. One caution is that I tended to run them for recreational round robins, so they are skewed to avoid having the best players and the weakest players on the court together. Causes more duplication, but more fun for the players.

Steve