On the decomposition of problems

There’s a joke that I can never quite get right, and even then telling it at a party often gets the rather unencouraging response that the tale attempts to illustrate. It’s all about how people reduce problems.

The Engineer, the Physicist, and the Mathematician are each given the task of solving the problem of the light bulbs burning out in the hallway of the science building. The Engineer sets about to design a longer lasting, high efficiency light bulb that will last much longer. Three years, they deliver this bulb to the science building and the custodian says they will try it the next time a light bulb burn out.

At the same time, the Physicist sets about to determine how to use the existing bulbs more effectively. They discover that the bulbs are running too hot, which cuts their life expectancy considerably. They design a new circuit would reduce the amperage to bring the light bulb within its apparent tolerances and extend its actual lifetime to the expected time. They deliver this after several months, much faster than the Engineer’s solution. No one does anything because the custodian is not going to change the existing wiring in the building since there is no burnt out light bulb at the moment.

Neither the Engineer nor the Physicist noted that there was no light bulb that was burnt out when they delivered their solution. Given the same problem, the Mathematician had gone into the hallway with a broom and broken the burnt-out lightbulb. Asked why they did that, they said they had noticed whenever the lightbulb was shattered and its glass had fallen on the floor, a new light bulb showed up. They could not explain how their solution worked for burnt out light bulbs, but they knew that they could restate that problem into one that had a solution.

Set aside the realms of reality in which each Archetype deals (from more concrete to more abstract) because they are all mostly doing the same thing: reducing most of a problem to something else. The first two, the Engineer and the Physicist, fall back on things they know: processes, formulas, and such. They have meat grinders that solve most of their problems, and they cram the problem into their particular meat grinder and crank the handle until their particular sort of sausage comes out.

The Mathematician, however, doesn’t have a meat grinder. They have an infinite number of meat grinders, so they find the way the one that takes their current problem. Having found that, someone else cranks the handle because, having found a solution, the Mathematician is on to something else. Perhaps a later Mathematician will find a shorter path to a better meat grinder, after which the Mathematics Department Chair will counsel them that re-solving known solutions is not a path to tenure.

And, as a postscript to that joke, the Physicist might see that and then break every lightbulb in the building to see what happens. But, only if they are an Experimental Physicist. Part of the magic of math is not showing how you did it.

There are two sorts of programmers: Those who get the joke and those who don’t. Or, there are 10 sorts of programmers: Those who understand binary and those who don’t. And so on.

I think about this joke when I solve other people’s problems or help them solve their problems. Often, I’m trying to figure out the class of problem they really have and at which level they need a solution. I can think like this because I’ve been practicing for decades, and in all of that practice, I’ve seen a lot of different situations.

The breadth of that set of situations is more valuable than particular skills. It’s not about the skills inherit in a problem. It’s the variety of solutions and scenarios that someone can bring to bear on the problem.

Think about that joke a little harder. At first it looks like it’s just making fun of the different dispositions of three closely related groups. But, the joke has a structure that’s close to universal: “same, same, different”. And, that structure is the point of the joke. By establishing two parallel scenarios, the joke gives a third so that you will apply the same thinking. The joke, by giving two exemplars, has forced you into a pattern of thought about the third. And, having done that, the joke can now depart from the frame it constructed. This allows the story to surprise you by breaking the frame.

Breaking the frame is the structure but also the literal narrative. You expect the Mathematician to do math thing, but they do human things. The Mathematician uses a solution that’s not particular to math even though that’s how it’s presented.

The Mathematician may have been thinking like a mathematician, but they used something outside of the realm of math (and available to anyone) to solve the problem. That is, their conception of the possible set of tools was infinitely wider because it wasn’t constrained by the occupational stereotype.

And yet, here I am, over half way through this, still using the same ideas. (“I apologize for such a long letter - I didn’t have time to write a short one.”, said Mark Twain, or Benjamin Franklin, or Abraham Lincoln, or Winston Churchill, or Blaise Pascal, or Henry David Thoreau, or Marcus Tullius Cicero Quote Investigator). All of this is to say that your ability to recognize the structure of the problem along with a big enough toolbox allows you to effectively solve problems.

Curiously, “Same, different” isn’t enough to make the joke because there’s no time to develop the pattern and lead the listener to the expected then denied conclusion. Then “Same, Same, Same, Different” seems like too much. But, some cultures are “one, two, many” while others are “one, two, three, many”, and maybe there’s “one, many”. So maybe their jokes are different.

Recalling slides

I was explaining to a friend that experienced poker players probably don’t do that much math. They might state odds and probabilities, but they probably have those memorized. They go straight from observation to final results because they have been in that situation so many times before.

That is, the more situations you’ve seen and the more ways you’ve experienced a problem, the more likely you can go straight to an idea of a solution. The more ways that you’ve previously approached a problem, the more likely you have an approach that will work for another problem.

In one line of work I’m involved in, the leaders talk about “slides”. You see something weird, and they say “add that to your slides”. Add that to the situations that you’ll remember. In some future scenario, recall the slides that fit and act. Sometimes that’s because these split-second, life-and-death scenarios don’t have time for reasoning. This is the basis for USAF Colonel John Boyd’s OODA loop in Organic Design for Command and Control:

orientation is an interactive process of many-sided implicit cross-referencing projections, empathies, correlations, and rejections that is shaped by and shapes the interplay of genetic heritage, cultural tradition, previous experiences, and unfolding circumstances.


Conversely, consider, Magnus Carlsen, the top chess player in the world (and perhaps of all time). His strategy is to present his opponents situations they have not seen so they cannot reduce his play to things they have seen before. He physically and mentally exhausts them by requiring them to think rather than fall back on prepared or known situations.

Curiously, part of the idea of the OODA loop is to create chaos for your opponent while forcing the other side to clarify their intentions. Robert Greene wrote:

The proper mindset is to let go a little, to allow some of the chaos to become part of his mental system, and to use it to his advantage by simply creating more chaos and confusion for the opponent.

Math is hiding your work

Consider Mark Jason Dominus’s post 484848 is excellent . He starts by proving for all n, 4bn - 7an = 4. Why? Because he’d already struggled with the problem and that was the answer. When he came back to answer the question, he took the knowledge he had at the end of his struggles and used it straightaway to answer the question. And, I then expanded Mark’s excellent number ideas greatly (excellentnums.com) before Matthew Arcus completely and brilliantly solved it with something neither of us had considered.

This is similar to something John Ousterhout says about clean, minimal code. We don’t know how long it took to get the code that simple and clean. We see end (sometimes intermediate) results.