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:

1. 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).
2. The average number of Repeat People will get gradually higher throughout the day. The first 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!

tl;dr: The mathematical definition of an idempotent function is subtly different to the definition used in software engineering. In software engineering an idempotent function is one that has the same impact on state, no matter how many times it is run. In mathematics, an idempotent function is one where f(x) =f(f(x)).

The concept of idempotence came up recently at work, in the context of infrastructure. A statement along the lines of “When treating infrastructure as code, it’s often important to ensure that your functionality is idempotent.”

I asked what “idempotent” meant, and I was given the (incorrect in all contexts) answer, “for the same input, you will always get the same output.” I was also told (correct for maths, but not for software) that the following function is not idempotent:

f(x) = x + 1

…but the following function is idempotent:

f(x) = x * 1

This instantly got me asking questions, because the first two definitions I was given were at odds with one another. The first definition describes a deterministic function. For instance: Both f(x) = x + 1 and f(x) = x * 1 are deterministic, ie given the same input, you’ll always get the same output (3 + 1 will always be equal to 4, and 3 * 1 will always equal 3).

But then I was given a better example, more relevant to the original conversation: If you have a function that adds an entry to a hosts file, you want to know that no matter how many times you execute that function, you will only ever add one entry to the hosts file. You don’t want to add more than one entry.

For instance, we start with a hosts file that looks like this:

127.0.0.1 localhost

We run our function, and now it looks like this:

127.0.0.1 localhost
255.255.255.255 broadcasthost

We run our function again, and nothing changes. The broadcasthost entry has already been added. Our function has nothing to do.

And then I found a true (in mathematics) definition of idempotence:

A function f(x) is idempotent if f(x) = f(f(x)).

To reiterate: The originator made a mistake: It is NOT true that idempotence is defined as “the same input always gives the same output”.

That is to say, if a function is idempotent and you apply that function to x, then you apply the function again to the return value, you still get the same result. Keep taking the result of each pass and sending it back into the function, and you still get the same result.

At this point I didn’t know about the difference between mathematical idempotence and software idempotence, and I was happy with my new definition: In our hosts file example, if our function takes the file content as an input and outputs the transformed result as an output, you can keep reapplying the function and you will keep getting the same result.

Using our new definition, we can easily see how f(x) = x + 1 is not idempotent, but f(x) = x * 1 is idempotent.

So far so good. But then I found this article claiming that pure functions are always idempotent, and my head exploded.

A pure function is one that has no side effects and no hidden state. The example given was this one:

f(a,b) = a + b

My confusion stemmed from two sources: Firstly, how can I apply my definition above – f(x) = f(f(x)) to this new example? It takes two parameters, but only returns one result! But secondly, how can it possibly be idempotent? It’s deterministic, yes, but any way you can find of repeatedly applying the same function to a new output will surely produce a different result? And what on earth does its pureness have to do with anything??

Well. I asked a bunch of clever people, and discovered that I had been dealing with the mathematical definition of idempotence, which is subtly different to the software engineering definition.

In software engineering, it’s all about state. My hosts file example was flawed because I had assumed that the hosts file content was being passed in as an input and then returned as an output. In fact, we are talking about a function that acts on the hosts file. This function’s input may be the hosts file path. Its output may be some kind of success code. It is not a pure function, because it will have the side effect (sometimes) of altering the state of the hosts file.

BUT our idempotent hosts-file-editing function can be run several times, and its effect on state will always be the same. No matter how many times we run this function, the hosts file will always be impacted in the same way.

The article that confused me so much was in fact making a very simple point: Pure functions are idempotent because they do not alter state. Therefore state is always impacted in the same way by multiple calls to a pure function, because it is simply not impacted. So in reality, most idempotent functions are not pure, but all pure functions are idempotent.

One common example of idempotence in software engineering is the HTTP specification – which states that GET, PUT and DELETE requests should all be idempotent, but POST should not.

At this point I will quote my colleague Mouad (and the Stormpath blog), who between them say this:

“The HTTP RFC have a better definition which goes:

A request method is considered “idempotent” if the intended effect on the server of multiple identical requests with that method is the same as the effect for a single such request.

The ‘intended effect’ as defined above is not the same thing as the returned value, example: calling PUT two times may return a different result in the second call (e.g. 409 conflict), but a PUT is still idempotent if the state and effect didn’t change by the second call, in other words, ‘HTTP idempotency only applies to server state – not client state’ ref. https://stormpath.com/blog/put-or-post.”

Hopefully your head has not exploded. Or if it did, I manage to unexplode it and return it to its former state. Hopefully also, no matter how many more times you read this post, your head will remain unexploded. And we have ourselves an idempotent blog post. Voila!