today I learned... | Enes Demirel

Sum of squared differences as a dot productJul 10, 2026July 10, 2026

Last week, I learned that you can represent the sum of squared differences equation with a vector dotted with itself ( $v^Tv$ ), instead of writing out the full summation.

Let's say we have:

\sum_{i=1}^{n} (x_i - y_i)^2 = (x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2

$x$ and $y$ are two different vectors. We want to look at their differences. Let's define a vector $v$ whose entries are those differences.

v = \begin{bmatrix} x_1 - y_1 \\ x_2 - y_2 \\ \vdots \\ x_n - y_n \end{bmatrix}

Therefore, the multiplication ( $v^Tv$ ) is equal to

v^T v = \begin{bmatrix} x_1 - y_1 & \cdots & x_n - y_n \end{bmatrix} \begin{bmatrix} x_1 - y_1 \\ \vdots \\ x_n - y_n \end{bmatrix} = \sum_{i=1}^{n} (x_i - y_i)^2

Steve Jobs & the Google logoFeb 21, 2026February 21, 2026

Did you know that, in 2008, Steve Jobs noticed that the Google logo displayed as an icon on the original iPhone did not match Google's official brand colors, spesifically the yellow in the second "o." He contacted Google to point out the issue. After reviewing the logo, Google changed the color to match the official brand colors. The change was only visible on the iPhone, and not on other devices.

Planets fit between Earth & MoonFeb 12, 2026February 12, 2026

Did you know that you can almost fit all the planets in the solar system between the Earth and the Moon by putting them next to each other without any space between them?

Wait, how?

It is actually pretty simple elemantary school mathematics.

\begin{array}{|c|c|} \hline \textbf{Planet} & \textbf{Mean Distance (km)} \\ \hline \mathit{Jupiter} & 142,984 \\ \hline \mathit{Saturn} & 120,536 \\ \hline \mathit{Uranus} & 51,118 \\ \hline \mathit{Neptune} & 49,528 \\ \hline \mathit{Venus} & 12,104 \\ \hline \mathit{Mars} & 6,792 \\ \hline \mathit{Mercury} & 4,879 \\ \hline \end{array} \\[0.5cm] \begin{array}{|c|c|} \hline \textbf{Total} & 387,942 \\ \hline \end{array}

Average Lunar Distance:

384,399 \text{ km}

Therefore, all the planets in solar system can almost fit in the space between Earth and the Moon.

A nice note: If you line the planets up top-to-bottom (pole-to-pole), they fit with plenty of room to spare!

The 32-bit air traffic control bugFeb 11, 2026February 11, 2026

Did you know that in 2014, a radar data processing system used by the regional air traffic control facility serving Southern California failed after running continuously for an extended period? The failure caused hundreds of flights to be delayed by 1 to 3 hours, and some were even canceled. Later, it was revealed that the failure happened because of a very small software bug involving a signed integer overflow.

But what is a signed integer overflow?

Computers store numbers in binary. Most older computers were/are still using 32-bit systems. That means that the biggest number a 32-bit unsigned integer can store is a binary value with 32 1's next to each other. That number is:

2^{32} - 1 = 4,294,967,295

However, computers need to store negative numbers as well. So, computer scientists in the 1960s came up with a solution. Using the most significant bit to store the sign. If it is 0, that is a positive number; if it is 1, that is a negative number. This is called a 32-bit signed integer. To be able to read a negative number, you have to invert all the bits and add 1.

Example:

\begin{aligned} &\rightarrow \text{ } 11111111 = -1 \\ &\rightarrow \text{ } 00000001 = 1 \end{aligned}

Therefore, we can now represent negative numbers, with a small caveat. The biggest positive number we can go up to now is 31 1's next to each other. So, our range became:

-2,147,483,648 \text{ to } 2,147,483,647

An integer overflow happens when a number exceeds the maximum value the data type can store. Which basically means that a 32-bit signed integer cannot exceed $2,147,483,647$ .

After the investigation, authorities found that the system was keeping track of time using a 32-bit signed integer, counting milliseconds. Which means the system was going to crash after:

2,147,483,647 \text{ ms} \approx 24.85 \text{ days}

They knew about this, but their solution was basically restarting the system. However, this time they forgot. This was a good lesson for those programmers not to keep track of time using a 32-bit signed integer.

A nice note: A 64-bit signed integer can run for $292$ million years (counting milliseconds) without crashing.

2^{63} - 1 = 9,223,372,036,854,775,807 \text{ ms} \approx 292,471,208 \text{ years}

I first read about this story in the book Humble Pi by Matt Parker. I strongly recommend it to everyone interested in math.

52! vs atoms in the solar systemFeb 10, 2026February 10, 2026

Did you know that all possible arangements of a 52 deck cards is greater than the number of atoms in the solar system?

Explanation:

We first need to calculate the total number of permutations of a 52 deck cards. This can be done with calculating how many ways we can arrange 52 distinct cards:

\begin{aligned} &\text{There are 52 choices for the first card.} \\ &\text{There are 51 choices for the second card.} \\ &\vdots \\ &\text{There is 1 choice for the 52nd card.} \end{aligned}

By the product principle, the total number of permutations is:

52 \times 51 \times \cdots \times 1 = 52! ≈ 10^{67}

The number of atoms in the solar system is estimated to be around $10^{50}$ .

52! > 10^{50}

Therefore, the number of permutations of a 52 deck cards is greater than the number of atoms in the solar system.