A math problem

I chanced upon a math problem. I didn’t read it somewhere, nor was it presented to me as a problem in so many words. What happened was this. I am taking a specialisation course in Business Strategy on Coursera. As part of each module, we have to submit a final assignment which is graded by three peers who are taking the course at the same time as me. In turn, I have to grade three assignments of my peers. When I got my completion certificate for this week, I noticed that my peer reviews were done by the same three people whom I had peer reviewed. This mortified me a bit. For while all three of the said peers had given me a perfect score in all the scoring questions, I had not done so for them. This makes the situation rather embarrassing and makes me look like quite a dick. Now in fairness, I had simply followed what the questions asked and in my (hopefully) fair and unbiased judgement, I had marked the scores according to my reading of their content. The only saving grace in all this mess is that your grade and peer grading results are not revealed to you until you complete your three reviews. So you have no idea who’s going to review your submission and what they might say. The downside is that I will probably find these same people in the next module and they are going to be none too pleased about evaluating the jerk from India who thinks he is superior. Or so I presume.

Now, all of this got me thinking. Why on earth did I get the same three people to review as they got my submission? Shouldn’t this sort of thing be avoided, for many reasons such as diversity of perspectives and a firewall against the very issue I’ve described above. The internet as a place is fraught with tricksters and fraudsters and no-good types. While you would expect people to remain level headed on a reputable platform like Coursera, this kind of assignment would certainly fuel mistrust and breed discontent. So here is the math problem to solve this dilemma:

What is the minimum number of people (N) needed to start a course if each person’s assignment has to be reviewed by three different people, each person must review assignments of three people and no person can review the assignment of a person who has reviewed theirs?

I am sure this problem has been pondered, analysed and even solved by Coursera themselves. No doubt also, that it may seem a trivial one to most of you. But I have a very linear way of thinking and math problems scare me. But this one I want to solve. I want to get as far ahead in to the solution as I can without resorting to googling the damned thing. So far, I’ve spent the night mulling it over and I’m going to try and establish a table to see if I can get a manual solution to the issue. Equations are coming to mind, but none that make sense right now. Unlike most exams we attempt in life (in Indian life at least), I don’t have a pattern to follow or syllabus to refer to. I have no clue what category of problem this is, whether I can actually solve it or what methods to apply to approach it. But I feel like trying.

I write this paragraph having spent half a day solving the thing. Here’s my solution, find fault with it if it pleases you.

Like I said, I have a very linear way of thinking. The very first formulaic thingy that came to mind was this:

Where N is the minimum viable number of people

While this makes sense, how then to proceed with our conditions? Set theory came to mind next:

I am not sure that this entirely covers our two conditions though

At this point, I thought a real world experiment might help much. I started imagining this as a problem involving my colleagues at work. If I were the first student, then how many people on the floor would be needed in order to get a closed set of people submitting and evaluating three assignments with no overlap? So I pushed aside the equations and made a simple table:

This took me a while, simple as it looks. I never claimed to be smart.

The reason it took me a while is because I started adding people very carefully, checking and double checking my work. Oh and also, I was actually working in an office at a job where I get paid a salary. So this stuff was done in between all of that, when my automation scripts were running and I would otherwise spend my time twiddling away at my phone. My colleagues of course thought that I was nuts.

Now we come to the second and last table which again took me quite longer than expected. Mostly because I made a simple error. I miscounted all the places person G was in the table above. Ah well.

Easy Peasy, as it turns out.

This then satisfies all given parameters of the problem. You need at least 7 and all further groups in multiples of 7. That’s rather unhelpful. So, if you need to start a MOOC platform, here’s a problem you’d probably never need a solution to.

The point of all this, I hear you ask? None whatsoever. I have no idea where this thought came from, though I can trace its exact origins and everything. I do not know what possessed me to do this. It is not the type of thing I usually attempt. I would perhaps think of such a thing, consider it a moment, not find an immediate answer and let it go. But I stuck with it and surprisingly found a solution (unless someone proves me wrong). It was a strange thing, completely out of character and I have no clue how I should feel about this. Perhaps it was a one off eccentricity you recount in overly loud stories to friends when you’re swimming in it up to your eyeballs. Perhaps it is something else. Perhaps it is nothing at all. We all have our little idiosyncrasies.