In his reflection on the pursuit of understanding, Oliver Wendell Holmes Sr. wrote, “I would not give a fig for the simplicity on this side of complexity, but I would give my life for the simplicity on the other side of complexity.” At first glance, this might seem cryptic, but it speaks to a truth we often encounter in learning, problem-solving, and life: real insight often lies beyond a confrontation with complexity—not before it.
To illustrate this idea, let’s examine a deceptively simple problem involving airplane seats.
The Airplane Seating Problem
Imagine $n$ passengers boarding a plane with exactly $n$ seats. Each passenger has an assigned seat, but the first passenger decides to sit in a random seat, not necessarily their own seat. Every other passenger behaves in the following way:
- If their assigned seat is available, they take it.
- Otherwise, they pick a random unoccupied seat.
The question is: What is the probability that the last passenger will sit in their assigned seat?
Engaging with Complexity: The Struggle
At first, this problem feels overwhelmingly complex. The first passenger’s random choice creates a cascade of randomness: each subsequent passenger’s decision depends on the choices made before them.
Here’s how I thought about this problem the first time I heard it: If the first passenger happens to take their own seat, then every one else will as well. That happens with probability ${1\over n}$. But what if they don’t? Well, suppose they occupy the seat for passenger $k$. Then passengers $2,3,…,k-1$ will take their seats but obviously passenger $k$ has to pick another seat, and so on. This leads to lots of different possibilities. Each scenario has a probability, and after a while things look too complicated.
Simplicity on this side
Faced with a situation like this, most people have a tendency to move on to something else, or say something like “everything here is so random that the answer has to be ${1\over n}$”. It is what one might say without engaging with the complexity of the problem. Holmes called such a response to be on this side of complexity.
Rules over moves
After struggling for a bit the fog begins to clear. The first passenger is special, and the others are all alike since they follow the same boarding rule. Let’s call them ordinary. When an ordinary passenger boards what can we say about the unoccupied seats they find?
Well, they can’t belong to any ordinary passenger who has already boarded, because the rule for ordinary passenger says that you have to take you seat if it is occupied. So the unoccupied seats fall into one or more of these categories:
- The seats of passengers who haven’t yet boarded.
- The seat of the boarding passenger
- The seat of the first passenger.
This leads to a cool realization: when the last passenger boards, the only unoccupied seat is either
- Their own seat.
- The first passenger’s seat.
We are almost done! Any passenger who boarded before the last one followed a rule that treats all of the unoccupied seats equally. So the probability that the last passenger finds their own seat unoccupied is ${1\over 2}$. In fact this is true of all regular passengers: the probability that they find their seat unoccupied is ${1\over 2}$.
The Journey to Simplicity
Initially, the airplane seating problem feels impenetrable because of its apparent complexity. But by carefully engaging with the rules and mechanics, we find the crucial insight. This is precisely what Holmes meant by “simplicity on the other side of complexity.”
On this side of complexity, simplicity is naive, and often completely wrong. It allows our biases, laziness and other factors to come into play. When we are on this side of complexity, we need to understand that it is wise to be skeptical of our analysis.
On the other side of complexity, it is often the case that while the struggle is hard, the reward is cool and the problem is so simplified that you know your conclusion has to be correct. In my own experience this approach works even better in real life than in math problems.