Math - Word Problem

February 12, 2009 2:53 PM

Math word problem inside. There must be a better way to figure this out.

A group of people want to do a "Dining Around" club. There are 12 couples. Each couple will host 1 dinner with 2 guest couples, and each couple will be a guest at 2 dinners, all on different dates. The goal is to not dine with the same couples more than once.

What we did: We wrote each couples' name on index cards 3 times. We placed 1 name card on the table for each couple as host. Then we shuffled around and randomly paired the remaining guest names with hosts. It took quite a long time to get it right.

Is there a formula (or better yet software, Excel perhaps) to make arranging easier?

posted by viva viola to Grab Bag (4 answers total)

This is similar to making a season play schedule, complicated by the fact that there are three teams on the field, so-to-speak. But it's also easier because you only need to do this for one "rotation" in a season.

Assign each couple a number from 1-12. List them out in three rows, then shift the second row to the right once. Since you don't want any repeat couples, shift the third row to the right twice. It would look like this:

  1  2  3  4  5  6  7  8  9 10 11 12   Hosts
 12  1  2  3  4  5  6  7  8  9 10 11   Guests
  9 10 11 12  1  2  3  4  5  6  7  8   Guests

So couple 1 hosts couple 12 and 9, and so on.
posted by muddgirl at 3:05 PM on February 12, 2009

Well, the third row is shifted to the right 3 places in my example, but that's beside the point.
posted by muddgirl at 3:06 PM on February 12, 2009

This is very similar to Kirkman's Schoolgirl Problem.
posted by junipero at 3:21 PM on February 12, 2009 [1 favorite]

Here's a slightly more generalized method.

Choose 3 numbers that add up to 12. As long as no two of the numbers are the same. So (2,3,7) works and so does (2,4,6). Call any 2 of these three numbers X and Y.

1) Number the couples from 1-12.
2) For each Host couple add guest (Host)+X. If Host+X is >12 then subtract 12.
3) Now add guest (Host)+X+Y. If this number is greater than 12 then subtract 12.
posted by vacapinta at 2:31 AM on February 13, 2009