The Beauty of Sharing: More Elegant Solutions
A few months ago, for the first time in my professional life, I shared an article I had written on LinkedIn. Nothing happened.
A few weeks ago, I did it again, and I was amazed by the result. Data and Computer Scientists from all over the world started interacting with the problem.
This article will explore the beauty of sharing, the good side of the internet, and more elegant mathematical solutions.
The article I shared shows a solution to the following problem, taken from this amazing book:
If you haven’t already read it, I would recommend it (link). Now, let’s get into it.
Thinking Graphically
My initial solution involved the summation of a geometric series. This sounds complex, but I tried to make the concept more accessible in my blog post.
A few minutes after I had posted this on LinkedIn, Samuele pointed out that the solution can also be found graphically.
It turns out that each black square is surrounded by two identical white squares. In other words, the black region is \(1/3\) of the pattern repeating indefinitely.
Thanks to his large audience and network, this started a series of reactions and comments.
A Solution from Another World
Then came the most elegant solution of all. A real UFO, something that I would never have considered. Guy mentioned that the area of the black region could be computed as follows:
\[ X = \frac{1}{4} + \frac{X}{4} \]
Wait, how?
Let’s pause for a second and let that sink in. Recursions are always confusing.
To make sense of this expression, let’s substitute \(X = \frac{1}{4} + \frac{X}{4}\) into the right-hand side of the expression shown above. We get:
\[ X = \frac{1}{4} + \frac{\frac{1}{4} + \frac{X}{4}}{4} = \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{X}{4} \]
Substituting \(X = \frac{1}{4} + \frac{X}{4}\) into the right-hand side of this expression again, we get:
\[ X = \frac{1}{4} + \frac{\frac{1}{4} + \frac{X}{4}}{4} = \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{\frac{1}{4} + \frac{X}{4}}{4} \]
Expanding further:
\[ X = \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot X \]
Repeating this process indefinitely yields the following expression:
\[ X = \frac{1}{4} + \frac{1}{4}^2 + \frac{1}{4}^3 + \dots \]
This is a geometric series that sums to \(1/3\), as shown in my original post.
The expression \(X = \frac{1}{4} + \frac{X}{4}\) has one advantage over the geometric series. It allows the following simple algebraic manipulations:
\[ X = \frac{1}{4} + \frac{X}{4} \]
Subtracting \(\frac{X}{4}\) from both sides:
\[ X - \frac{X}{4} = \frac{1}{4} \]
Simplifying:
\[ \frac{3}{4}X = \frac{1}{4} \]
Multiplying both sides by \(\frac{4}{3}\):
\[ X = \frac{1}{3} \]
This is how Guy came up with an answer in four lines, where my proposed solution took a few paragraphs.
If you think of yet another way to solve this problem, feel free to share it in the comments below.
What to Do from There?
Keep sharing! I am naturally a reader and an abstract thinker. I learn most from text and symbolic manipulations. Others are more visual or practical thinkers. It is by exchanging ideas that we get to understand the world through our many perspectives. Isn’t this exciting?
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