Math Homework

Math Homework

# I have 8 groups and 8 games. Each game is played by 2

Customer Question
groups. How do i arrange...
I have 8 groups and 8 games. Each game is played by 2 groups. How do i arrange when the groups go to which games so that each group played each game once and each group played with another group at least once.
Submitted: 2 years ago.Category: Math Homework
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9/2/2015
Math Tutor or Teacher: Ryan, Engineer replied 2 years ago
Ryan, Engineer
Category: Math Homework
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Hi,

Thank you for using the site. I'll be happy to try to work this out for you.

Scheduling problems like this are often more complicated than one would expect. In this particular case, as I understand it, there are eight different games to be played. Since there are only eight groups, and each game requires two groups playing against each other, only four of the games can be played at any one time. This complicates matters, as it becomes necessary to make sure that each group has played each game once.

I think I can work something out, but I won't know for certain until after I have actually done it. I'll keep you posted though.

Thanks,

Ryan

Customer reply replied 2 years ago
Alright thank you. I know that. Thats why it is difficult. Looking forward to your answer soon. Thank you.
Math Tutor or Teacher: Ryan, Engineer replied 2 years ago

Hi,

I'm working on it. I have the pairings of groups. I just need to get them assigned to games in the proper order so that each group plays each game once. It's not really a trial-and-error kind of problem, so I may need to write a computer program to check possible combinations exhaustively, and that will take some time to accomplish. I'll let you know as soon as I have something.

Ryan

Math Tutor or Teacher: Ryan, Engineer replied 2 years ago

Hi again,

I'm sorry to report that after working on this for several hours, I haven't yet been able to distribute the group pairings among the eight games. I'm not giving up yet, but I wanted to let you know that this might take a while to accomplish. Even if I get a program running, it may take a considerable amount of time to run due to the possible number of combinations that it would have to check. Was there a particular deadline that you had in mind for this?

Ryan

Customer reply replied 2 years ago
HelloThank you for taking your time to do this. Probably by next week. I just want to be sure that up till now that no charges has been charge.
Math Tutor or Teacher: Ryan, Engineer replied 2 years ago

Hi,

Ok, good to know. I'll get back to this later (staring at it some more isn't going to help right at the moment).

Regarding your question about any charges, I would recommend that you contact a Customer Service representative about that. I'm not an employee of the site, so I don't have any access to your billing information, nor do I have any authorization to discuss any of that. I'm limited to answering the math questions.

There should be a link on the page that you can use to contact them for information, or you can call them at(###) ###-####

Ryan

Math Tutor or Teacher: Ryan, Engineer replied 2 years ago

Hi,

I spent a considerable amount of time on this Wednesday night and Thursday, but to little avail.

Obtaining the pairings such that each group plays a game with every other group once (which would be a total of seven games), and then one repeat (to make the eighth game), is easy to do. The problem is in scheduling the match-ups so that each group plays each of the eight games once.

The computer program that I wrote works, but there are just too many possible combinations to check. Each row of the schedule has 40,320 possible permutations of four match-ups and four empty slots (to account for the eight different games). With eight rows in the table, there are a total of (40320)^8 possible arrangements. The trick is finding the ONE (or one of the few) arrangements that meets the requirements of having each group play each game once.

The program was going through over 4 million arrangements every minute or so. However, due to the huge number of possibilities to check, it would have literally taken 3.3 x 10^24 YEARS to run through all of the arrangements. I humbly submit that this is more time than I can afford to spend on this problem. :)

My next attempt was to try to break the problem into smaller pieces. I thought that if I made a schedule for FOUR games, using half of the pairings, and then did the same with the other half of the pairings, that I would be able to combine them into a single schedule. Or, perhaps, you could have used it in two phases. For instance, a morning session and an afternoon session, each using only four of the eight games. However, this didn't work out either as no solution was found after exhausting all possibilities. (These are much smaller problems, so the computer run time less than a minute.)

I then attempted to search the Internet for possible solutions. As I mentioned earlier, this type of problem arises frequently in tournament scheduling, particularly in golf. As a result, there are many problems that are already solved. Unfortunately, I was not able to find a solution that matched your particular scenario, nor was I able to find anything that was close enough to modify.

All of these attempts assume that there is only one copy of each of the eight games available. (I don't think you mentioned whether these were board games, card games, or something else.)

A simpler approach might be to have multiple copies of the games. For example, if you had four copies of each game, then you could have all eight groups playing the same game in each round. Then, after eight rounds, every group has played a game against every other group, and every group has played every game.

If you wanted to do that, the pairings for each round are as follows:

Round 1: AH BG CF DE

Round 2: AF BE CD GH
Round 3: AD BC EG FH
Round 4: AB CG DF EH
Round 5: AG BF CE DH
Round 6: AE BD CH GF
Round 7: AC BH DG EF
Round 8: AH BG CF DE

I hope this helps. If there is anything else that I can do to help, please do not hesitate to ask.

Thanks,

Ryan

Customer reply replied 2 years ago
Hello,As my games are station games and i have 8 groups. In order to make the team get a chance to play with others thats why it is to play with other team at least once. But its alright since you cant answer my question. But thanks for taking your time to try.Thank you
Math Tutor or Teacher: Ryan, Engineer replied 2 years ago

You're welcome.

Ryan, Engineer
Category: Math Homework
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