“Simplicity,” said Leonardo da Vinci, “is the ultimate sophistication.” The virtues of simplicity have been echoed by many of the greats: Einstein, Shannon and even Walt Whitman (“Simplicity is the glory of expression.“) .
But this raises the important question : exactly how does one simplify? Simplicity, it turns out, is much easier said than done in science, math or life!
Zoom in: Look for the smallest instance
It is almost always worth reducing a complicated problem to its simplest case, and seeing if that yields insight.
Consider the early experiments in genetics by the nineteen century monk, Gregor Mendel. His goal was nothing less than to understand how biological traits are inherited. But he simplified the problem to the breeding of pea plants, which he could conveniently grow. Rather than attempting to understand all the traits of the plant simultaneously, Mendel focused on a single trait at a time—such as flower color or seed shape. By isolating and simplifying his study, he discovered fundamental patterns of inheritance, which eventually became the basis for modern genetics. This reduction to a simple instance allowed Mendel to develop the laws of heredity, which could then be generalized to more complex combinations of traits.
Now lets apply this powerful principle to the following problem:
Two men are at seated at a rectangular table and play the following game. The first player places a penny anywhere on the table so no part of it is off the table. The second player does the same without covering any part of the first penny. This continues until a player cannot place a penny on the table. This player loses. Who wins the game if both players play optimally?
At first, this problem appears to be strange. Doesn’t the strategy depend on the size of the pennies relative to the dimensions of the table? Also, since the pennies can be placed in patterns that leave “holes” that are too small to fit a penny, maybe the strategy involves creating these holes to trap the opponent. How to proceed?
Here’s where simplification comes to the rescue! What happens when the table can fit only one penny? Then the first player would occupy the center and win. Now make that table big enough to support a second penny, given that one is in the center. Since the table is rectangular, the position taken by the second penny has a mirror image for the first player to make his second move. To clarify, draw $x$ and $y$ axes on the table with the center as the origin. If the second play places the center of their penny on the point $(x,y)$, the first player places a penny with its center on $(-x,-y)$. Now we see the strategy. The first player places his first penny in the center and then mirrors the moves of the second player to guarantee their win. Cool, right?
Zoom out: Look for a more General Problem
But the trick of finding the simplest instance of a problem does not always work. In fact, abstracting the problem, or making it more general often helps. How can one simplify by generalizing?
The problem of optimizing a city’s bus routes might seem daunting when viewed in isolation. However, by recognizing it as a specific instance of the “network optimization” problem—a general class of problems studied extensively in fields like mathematics and computer science—planners can apply existing algorithms and methodologies. Viewing the problem in this generalized context not only simplifies it but often reveals solutions that are both efficient and scalable.
The development of vaccines for new diseases, like COVID-19, often benefits from generalizing the problem to the study of immune responses across a range of pathogens. By applying what was learned from similar viruses, researchers developed an effective COVID-19 vaccine in record time.
The Dance of Zooming In and Out
The most effective problem solvers often alternate between these two modes—zooming in to simplify and zooming out to generalize. When designing a building, architects might zoom out to understand the structure’s role in the urban landscape—its aesthetics, environmental impact, and interaction with surrounding buildings. They then zoom in to consider specific details, such as material selection or load-bearing calculations, to ensure functionality and durability.
As a final example, consider this truly instructive example of zooming in and out from Geometry. I learned this when I was in my early 20’s and was reading “Mathematics and Plausible Reasoning”, by the great mathematician and teacher Georg Polya.
Suppose we want to prove the Pythagorean Theorem:
Given a right angled triangle with sides $a$, $b$ and hypotenuse $c$, $$c^2=a^2+b^2$$ $~$
One way to think about this result is to draw squares of lengths $a$, $b$ and $c$ and to say that the area of the larger square is the sum of two smaller ones.
Now let’s generalize. Instead of constructing squares, let the shape on each side be any polygon, as long as the polygons are similar (scaled versions of each other) .
In the figure above the shape is a pentagon. In this general problem we want to show that the sum of the areas of two smaller polygons (in this case we illustrated with pentagons) is equal to the area larger polygon. Suppose the smallest pentagon has area $A$. Then by choosing $\lambda = {A\over a^2}$ we can write its area as $\lambda a^2$. Since the other two pentagons are scaled versions, their areas will be $\lambda b^2$ and $\lambda c^2$. This is the same as showing that
$$\lambda a^2+\lambda b^2 = \lambda c^2.$$
But this looks trivial and somewhat pointless since we can always cancel out the $\lambda$’s and get back the original problem. The two problems are equivalent! And we don’t know how to solve the original problem so how could multiplying both sides of the equation by $\lambda$ help us? But what if we pick a special shape for which the sum of the two smaller areas obviously equals the third one?
The left figure below shows the the three polygons when the polygon is a scaled version of the original triangle itself.
The triangle on the hypotenuse is a reflected version of the original triangle and so has the same area as the triangle. And the other other two triangles can be “folded in” to correspond to the triangles ABD and BDC in the right figure. So the sum of the areas of the smaller triangles is equal to the area of the larger one!
For this special case the result obviously holds. There must be some constant $\lambda_T$ such that these triangles have areas $\lambda_T a^2, \lambda_Tb^2$ and $\lambda_T c^2$. Therefore:
$$\lambda_Ta^2+\lambda_Tb^2 = \lambda_T c^2,$$
and by cancelling $\lambda_T$ we have proven the Pythagorean Theorem. This wonderful proof is actually due to Euclid himself.
It’s worth recapping what just happened. We started with a difficult problem involving squares and turned that into a problem of the areas of similar polygons. We observed that this general problem is equivalent to the original problem, and in fact is equivalent to finding any particular polygon for which the result is obviously true. We found such a polygon: the original triangle itself, and thereby proved the original result.
After figuring all of this out, it is possible to rewrite the proof without even referring to the general problem. Polya, however, wanted to emphasize the role of generalization in the process of arriving at the solution. It was as if he was revealing a great magician’s trick.
Simplification as a Discipline
Simplifying a complicated problem isn’t just a technique; it’s a mindset. Before jumping to the simplest case, see if the problem lies in a more general class. Then, look for the simplest case that comes closest to solving the original problem. It may take a few iterations of zooming out and in, but more often than not, you will be on the path of finding that elusive simple solution.