The other day I posted this problem.

In the process of chatting about it, an associated mathematics problem came up, about permutations and probabilities.

To remind you of the basic problem: We have 450 conference attendees. We have one day of workshops. There are five sessions throughout the day, and five workshops to attend. We want every attendee to attend every workshop. The workshops are being repeated throughout the day: In every session, there are 10 rooms available. There are five workshops, and each workshop is duplicated. So for instance, workshop 1 will be run in rooms A and B, workshop 2 in rooms C and D, etc. They are repeated throughout the day, so one attendee might do them in the order 15423, and another might do 42315.

There will be 45 people in each workshop. The nub of the problem is that the organisers want as much cross-pollenation as possible. People do not get to choose what order they do the workshops: they are told which room to be in at what time. The organisers would like each attendee to meet *as many new people as possible* in every single session.

So here’s the maths question: What is the minimum average number of repeat people that attendees have to meet in each session? By “repeat people” I mean “people they have already attended a previous session with.”

There is one extra constraint: There is one group of 30 people (labelled “FRIENDS” in the spreadsheet below) that have to be kept together in every session. So they only get 15 people in every session that they may not have met before, ie for them the min value of Repeat People is 30 in every session. This also has an impact on all the other people in the other sessions.

However this is still an interesting problem even without the extra 30-people constraint.

So in session one, Repeat People = 0. In every one of the workshops in session 1, attendees are in a brand new group of people that they have never *attended this conference with* before.

So what is the minimum average value of Repeat People for session 2? I’m specifying average, because it would be possible to keep things pure for at least one attendee, where they meet 44 new people in every single session. But that would have an adverse effect on everyone else. So I’m aiming for everyone to have roughly the same number of Repeat People per workshop.

I have two theories, one of which I can prove:

- Apart from in Session 1, the average Repeat People is definitely greater than zero. You can’t avoid meeting attendees more than once (if you’re spreading the load evenly) (I can prove this one).
- The average number of Repeat People will get gradually higher throughout the day. The firstĀ solution I came up with maintains a constant number for Repeat People after session 1, but it’s non-optimal (I think).

Here is a way of visualising the problem:

I’ll be very interested to hear any solutions!