If you’re organising a conference or suchlike, you often want to give people a chance to get to know each other. You could pour them into a big room, pour them some drinks, and leave them to it. But as a way of fostering new connections, this doesn’t really work: people gravitate towards people they already know, and they tend to stay in the groups they started off in. And many people, particularly newcomers, find this sort of event awkward and uncomfortable.

I can provide materials for you to run a different and more effective kind of networking event. It’s like speed dating, but with people meeting in small groups rather than in pairs. Typically, people meet in groups of 4 or 5, and are given five minutes for everyone to introduce themselves to the group (a minute each). Then the groups break up, and people gather into new groups and introduce themselves to their new group. This keeps going for up to about 10 rounds, depending upon the number of people at the event.

The key feature of my materials is that they guarantee that nobody meets anybody else twice. A number of labelled meeting stations are set up in the room, and everybody is given a sheet which tells them which station to go to on which round. Here’s an example: This event has 7 rounds, after which everyone will have met 28 different people, and will have met nobody twice. This is much more efficient than free-format mingling, and the structure makes it a lot more comfortable for people who find such mingling difficult.

You could of course have your groups do whatever you want for as long as you want. What I provide is a structure that mixes people up efficiently from round to round.

If you have 9, 15, 16, or 25 people, we can arrange things so that everyone meets *everyone* else, exactly once.

If you’d like to run one of these speed networking events, please get in touch.

#### You may be thinking…

*I’m sure I could work a system out for myself. Do I really need your help? * That’s what I thought when I tried to organise my first speed networking event. It turns out that it’s a lot trickier than it looks: no-one has discovered a systematic way of shuffling people that stops people meeting twice. It’s actually a famous problem. See below for more details.

*A minute isn’t enough to get to know someone. *True, but that isn’t what we’re trying to do. The idea is to meet as many people as possible, so that people know who might be worth seeking out later. People also get an easy opening gambit to break awkward silences in the lunch queue: “Didn’t we meet in the networking event?”. There’s also nothing to stop you changing the timing or changing what people do when they meet in their groups.

*It sounds regimented and ghastly. I’d hate it! *A few people think this, and then change their minds when they have a go. And if you’re at an event where you know hardly anyone, the structure is really helpful.

*Why can’t everybody always meet everybody else? *This is the ideal, and for 9, 15, 16 and 25 people, it can be done. But for most group sizes, it isn’t possible, even in principle. The best that can be done is for no-one to meet anybody else twice. There are more details later if you’re interested.

#### How it’s done, *or *The Rev. Kirkman and his Problem

When I first tried to arrange one of these events, I confidently sat down at my kitchen table with a pen and paper, naively believing that there would be a simple way of shuffling people between groups that avoided people meeting each other twice. A couple of hours later, I was in despair, surrounded by unsuccessful scribblings. I couldn’t do it, even for groups much smaller than the 100 people that I had in mind.

Now if you can’t do something, there’s nothing so reassuring as the discovery that nobody else can do it either. A little research showed that I was trying to solve an instance of something called the Social Golfer Problem, which itself is a generalisation of the delightfully named Kirkman’s Schoolgirl Problem.

Thomas Penyngton Kirkman was a Victorian clergyman and mathematician of some stature. In 1850 he contributed the following problem to a magazine entitled *The Lady’s and Gentlemans’ Diary: *

Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.

This is a particular case of the problem that I was trying to solve, with 15 people meeting in groups of 3.

It turns out that nobody has discovered a systematic procedure for generating solutions to problems of this kind, which excuses my failure to do it myself. I then turned to less systematic methods, in which I evolved solutions to the problem by brute force application of my computer.

Briefly, my program starts off by allocating people randomly to groups on each round. Clearly, many people will meet each other twice. The program then tries small random adjustments to the group allocations. If an adjustment improves matters (fewer people meet twice) it’s kept, and if it makes things worse, it’s rejected. This happens millions and millions of times.

In some cases, the procedure reaches a successful solution where no-one meets anyone else twice. In others, it reaches a local optimum: an arrangement that isn’t perfect, but which can’t be improved by any small change. If this happens, it starts again with a new entirely random allocation of people and has another go. It repeats this indefinitely. In some cases, it manages to find a solution eventually, but in others, it becomes clear that, if there is a solution, my procedure isn’t going to find it. The greater the number of rounds in the event, the less likely it is that a solution will be found.

#### Can everyone meet everyone else?

Such a “perfect” event is possible only for events with certain specific numbers of people. Definitely possible are 9, 15, 16, and 25 people. There are other numbers of people which may be possible, but my process can’t find solutions, and there are further numbers of people where I know that it’s not possible even in principle.

Suppose we have *N* people in our event, meeting in groups of size *G*. There are various relations between *N* and *G* that have to hold if we are to create a perfect event.

For any event, even non-perfect ones, it’s necessary that *N* is divisible by *G*; this is so that we can split the people into groups of size *G*.

Also necessary for any event is that the number of people in each group (*G*) must not exceed the number of groups (*N*/*G*). If this is not so, people are bound to start meeting each other again on only the second round.

For a perfect event, there’s a further constraint. On each round of the event, each person meets *G*-1 people (everyone in their group except themselves). If, over the whole event, they are to meet all *N*-1 other people once only, *G*-1 per round, then it means that *N*-1 must be divisible by *G*-1.

So for a perfect event with *N* people, we must be able to find a group size *G* such that

*N*is divisible by*G**G*≤ (*N*/*G*)*N*-1 is divisible by*G*-1

There are many pairs of *N* and *G* that satisfy these rules. Here are the first ten, in the format *N*(*G*):

9(3), 15(3), 16(4), 21(3), 25(5), 27(3), 28(4), 33(3), 36(6), 39(3)…

Of these, my system solves 9, 15, 16, and 25 (these solutions exist elsewhere – for example 15 is Kirkman’s problem, for which several distinct solutions are known).

For all the other numbers that I’ve tried, my process doesn’t even get close to finding a perfect solution. I don’t know whether these cases are in principle impossible (because there are other contraints on *N* and *G* that I haven’t spotted), or whether they are in principle possible but my process can’t find them.