# Can 12 People Share 3 Tables in 5 Days for Dinner?

• Xamfy19
In summary, there are 169 possible solutions to seating arrangements in 5 days for a dining room with 3 tables. Each table can seat 4 people, so each person has a partner to sit with them at dinner.
Xamfy19
Hello All;
12 people use a dining room with 3 tables. Each table can sit 4 person. How to arrnage the seatings in 5 days so that each person can share the table at dinner with each other person?

thanks alot,,,

A general approach doesn't spring to mind but notice that if you arrange the people into pairs, each pair has five other pairs to sit with (i.e. one pair each day). Say you pair them (1, 2), (3, 4), (5, 6), ... (11, 12).
$$\begin{array}{|c|c|c|}\hline \mbox{Day}&(1, 2)\ \mbox{sits with}&(3, 4)\ \mbox{sits with} \\ \hline 1&(3, 4)&(1, 2) \\ \hline 2&(5, 6)&(9, 10) \\ \hline 3&(7, 8)&(11, 12) \\ \hline 4&(9, 10)&(7, 8) \\ \hline 5&(11, 12)&(5, 6) \\ \hline \end{array}$$
You can just complete the table by hand. You may be after a better way to do this, but it does work ;)

That's quite a good way. ANother way to explain it is, after rewriting it as a problem with 6 people to meet on five days (thinking of a pair as a person) sit them down as pairs

1 2

3 4

5 6

then keeping 1 fixed rotate the other five pairs anticlockwise (or clockwise) once each of the 5 days.

This looks very much like a problem adapted from Bridge seeing as it can be done with pairs.

So I can't help but wonder how many solutions there are (for pairings). 6C2 = 15 unordered pairs. But how do you split those up into 5 sets of 3, all disjoint?

At the start of each round of choosing 3 pairs, each individual (i.e. 1, 2, 3, 4, 5, 6) is in n of the p pairs. For the first round n = 5 and p = 15, and each round you subtract 1 from n and 3 from p. Figuring out how each round of choosing works is trickier; I see what's at work, but it's not clear how it's working. Every time you choose a pair, all pairs containing those individuals cannot be chosen that round, thus for every other individual, 2 pairs containing it are excluded. However, the choices are further restricted in a way I can't really explain. For instance, say {1, 4} is the first choice in round 3, leaving {2, 3}, {2, 6}, {3, 5}, {3, 6} -but I can only choose {2, 6} or {3, 5}.
Anyway, working it through, I get:
Round 1: 15C1 * 6C1 * 1C1 = 90
2: 12 * 4 * 1 = 48
3: 9 * 2 * 1 = 18
4: 6 * 2 * 1 = 12
5: 1
And I think I just sum the rounds: 169 solutions.? That seems like too many.

Last edited:
thanks alot, gentlemen!

## 1. How should I position my body when sitting to meet everyone?

When sitting to meet everyone, it is important to maintain an open and inviting posture. This means sitting up straight with your shoulders back and your chest open. Avoid crossing your arms or legs, as this can make you appear closed off and unapproachable.

## 2. Should I make eye contact with everyone while sitting?

Making eye contact is a crucial aspect of effective communication, so it is important to make eye contact with each person you are meeting. However, be sure to also break eye contact periodically to avoid staring or making anyone feel uncomfortable.

## 3. Is it necessary to introduce myself to everyone while sitting?

Introducing yourself to everyone is not only polite, but it can also help to establish a friendly and welcoming atmosphere. If you are in a group setting, it is a good idea to go around and introduce yourself to each person individually.

## 4. How should I position my hands while sitting to meet everyone?

When sitting to meet everyone, it is best to keep your hands relaxed and in a comfortable position. You can place them in your lap or rest them on the arms of your chair. Avoid fidgeting or playing with objects, as this can be distracting.

## 5. Is it important to smile while sitting to meet everyone?

Smiling is a simple but powerful way to create a positive and welcoming impression. While sitting to meet everyone, be sure to smile genuinely and naturally to show that you are happy to be there and open to meeting new people.

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