The Mathematical Theory of Gay Speed Dating
Speed dating addresses an age-old problem: given that your ideal partner is somewhere on this planet, how on earth are you supposed to locate him? He might be the one dancing right next to you at a party - but you wouldn't know, if you didn't strike up a conversation. Which may never happen, if you're shy, or the music's too loud, or for all kinds of reasons. So maybe you venture out on blind dates set up by mutual friends. But what if you realise in the first few seconds that there's no chemistry (and not even a hint of physics) between the two of you? Too late to back out. You have to see a painful evening through and then, perhaps, fend off phone calls for some days.
The speed-dating solution is to wrap the whole thing in the sanitised cloak of procedure. Get a number of people (assumed to have some common background) to pair up with each other in as many combinations as possible, in an organised setting. Each pair talks for a fixed and very short time, say three minutes. If both members indicate an interest in meeting again, the organisers later give them each other's contact details. Thus a speed date can lead to a comfortable and leisurely second date where each party is aware that the other has expressed some interest. Studies indicate that the first few minutes of conversation play a key role in determining if a relationship is likely to blossom, so speed dating really is the most effective way to start off the search for Mr (or Ms) Right.
The actual procedure (or "algorithm") must be powerful enough to maximise the number of partners each participant meets, but simple enough to minimise confusion. As we'll see, the optimal solution depends on whether the dating group is straight or gay. And it involves some nifty mathematics.
A "straight" speed dating group is, by definition, one whose members occur in two varieties - let's call them "females and "males"- who are interested only in their opposites. Speaking abstractly, this need not involve any heterosexuality - for example, the group may consist of gay men self-identified as "top" or "bottom". It's quite the same problem to a mathematician. What's important is the notion of "parity" or "grading", a label that can take two different values (like "male" and "female", or "top" and "bottom", but the actual names don't matter). For straight dating, we need an algorithm that preserves the grading, so that one type consistently meets the other.
A nice solution is the "bicycle chain". Members are lined up in two facing rows of chairs, with males and females alternating in both rows. At the start, every male is facing a female. Each one converses with his or her opposite number for the specified three minutes. Then a whistle blows and everyone moves to the chair on their right. This leaves one facing the person two places to the left of one's previous partner. Therefore the "grading" is preserved: if your previous partner was a woman, the next one will be a woman too. Besides being simple, "bicycle chain" is effective, because by the end of it every male in the group has dated every female and vice versa.
Here is an illustration with four males M1,M2,M3,M4 and four females F1,F2,F3,F4:
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Focus on one female, say F3. The diagram shows that she meets first M2, then M3, then M4 and finally M1. Thus by the end she has dated all the males in the group.
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When we decided to organise gay speed dating, it emerged - as usual at the last moment - that the required algorithm was not so simple. Apparently for the first time ever, we would be trying to get people without a grading to speed-date (and it's noteworthy that few urban gay men in India even identify as "top" or "bottom"). "Bicycle chain" does not work, precisely because it preserves the grading. Thus in a group of 20, it would let each guy meet only 10 others, as against the desired 19. Tinkering with the algorithm can increase the number of partners from 10 to 15 - good, but not good enough. Was there any simple procedure to get each of 20 men to speed-date the remaining 19? Or is mathematics hetero-normative, terrifying thought?
The perfect solution does exist, and we found it in the nick of time. It is a variant of the "bicycle chain" that is as elegant as it is baffling. Start with two rows of people seated on facing chairs. But now, ask one person (anyone) to remain fixed in his seat, while the others rotate according to the bicycle chain. As it moves, the chain bypasses the fixed person as if he were not there. It's easy to see that the fixed person ends up meeting all the others, since they occupy the chair opposite him one by one, while he never moves. What is more remarkable is that all the other members of the group also manage to meet each other precisely once. And the procedure is both simple and practical - as soon as we discovered this solution we implemented it, and it worked like a dream.
This is illustrated in the following figure, where M8 remains fixed:
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Focusing on, say, M3, we see that he meets M6, M1, M8, M5, M7, M2 and M4 in that order. And never meets anyone twice.
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The mathematical basis of this problem has to do with something called the "permutation group". Hardly everyone's cup of tea, a "group" in mathematics can be thought of as a set of transformations on a collection of objects. The transformations are required to possess properties that allow them to be studied abstractly, for example they can be combined with each other. The permutation group is made up of all ways of collecting N objects into smaller subsets. Because dating involves two people, we start with an even number of objects and collect them into pairs (but one could also incorporate futuristic sexual trends where people choose to date in groups of three or more!). Now, pairs form a "conjugacy class" of the group (appropriate terminology, I suppose, for something that's supposed to lead to conjugal bliss). All that's left is to permute these permutations so that every pair shows up once and only once!
That still leaves a few steps to be worked out, but this was not really meant to be a tour of group theory, so we'll skip them. And it has to be admitted that simple trial and error, with no use of formal mathematics, can - and actually did, in the present case - lead to the solution. But that's the beauty of recreational mathematics - you can attack the same problem with or without formal training, and your chances of success are usually the same either way. Which reminds me of another famous puzzle, also connected both to permutations and to sex. Consider two men, two women and two condoms. Each man needs to, umm, copulate with each woman, using only the two condoms, and without anyone infecting anyone else. The condoms cannot be washed. So, as one might put it, what's the fucking algorithm? There's a simple one (try this by yourself first, if you like):
But wait, that's not the end. Imagine generalising this problem to the case of N men, N women and N condoms. At a venerable institute of mathematics in Europe, a young British mathematician once stood seriously before a blackboard in the corridor, chalk in hand, attempting a solution. Oblivious to the passing faculty, students and staff, he was heard muttering to himself: "Now let's see, the first man fucks the second woman, then he puts on the other condom and fucks the third woman and then the second man fucks the first woman and after that she is fucked by..." Despite the amusement he caused, he was, sadly, unable to solve the problem, and I believe it remains unsolved to this day.
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