Think Like Gauss by Visualizing This Problem

Added: 2026-05-19

[00:00:00]Here's a classic question that you probably heard of before. It is simple, but we will look at it from a completely different perspective that will show whether you know how to think about mathematics. What is 1 + 2 + 3 plus and so on up to 200?

[00:00:14]Now, stop right there. How long do you think you would take to calculate the answer in your mind? A variation of this question, which was finding the sum of the first 100 positive integers, was first asked to the young Frederick Gaus when he was a school boy. His teacher was hoping it might occupy him for a nice half an hour, but unfortunately for him, Gaus managed to calculate the answer in seconds in his head.

[00:00:39]Of course, we have no idea whether this story actually happened to Gaus, but here's the important thing. I don't know how long it took to find the answer or even if you calculated it at all, but there's a famous trick to find the answer. In case you didn't know, the trick consists of adding the first with the last number, which gives us 201.

[00:00:57]Then we add the second with the second to last or 199. This gives us 201 once again. And if we continue this process, we always get 201.

[00:01:08]But the question now is how many 201s will we get at the end? The answer is half of this total length. So 100. Since we need to add 201 100 times, we can simply multiply 100 * 201. And then we get 20,100.

[00:01:28]This is a great question, but there is one danger in the way it is usually taught. It can make mathematics look like nothing but a bunch of tricks. As though the difference between understanding or not understanding something is whether somebody already showed you the the trick before. A good question is if you didn't know this trick, how would you find the answer? In mathematics, relying on tricks or in in advanced mathematics, relying on predetermined methods or logical steps, but not fully understanding why exactly you're using them is something very dangerous because it's just a matter of time before you get stuck. Because the point is that you don't really want to know the sum of 200 numbers. The number 20,100 is meaningless by itself. What you really want to know is how to think like Gaus thinks. And to think like Gaus, we have to think of bananas. And by the way, this false belief of relying on tricks in mathematics and also the discussions in this video, they all come from this book called Mathematica by David Bessis. And let us know in the comment section if you guys agree with his point of view. So the easiest way to see why mathematics is not just logical steps is to think about a recipe for banana muffins. The word I want you to focus on here is bananas. You know exactly what a banana is. I don't need to draw it for you. and you know that if it's really brown, you probably should throw it away. I don't have to tell you to peel it. You're not going to eat it this way. But if you've never seen a banana before, these things are not very obvious. And the exact same thing is true for the problem we started the video with. When Gaus was thinking about the problem, he was so familiar with how he needed to visualize the numbers that no one needed to tell him. It was as obvious to him as bananas are obvious to us. And it all comes down to being familiar with mathematical concepts down to your core. That's why we strongly believe that starting with intuition is the best approach to learn mathematics.

[00:03:19]Even though rigger later on needs to complement the intuition that you built before. And by the way, if you're serious about learning mathematics and expanding your knowledge with an intuitive approach, check out our catalog of PDFs. Each of them comes with a YouTube video and they are built on intuition, concrete examples, rigorous explanations and only then exercises with detailed solutions.

[00:03:43]Check them out in the description of this video.

[00:03:47]So in mathematics, the answer is not to just blindly follow the logical steps of a proof. If you don't understand the big idea, like the big picture of it, of of what you are trying to accomplish, but just try to memorize shortcuts, you won't really understand what the theorem or or the result you're studying is all about, and you'll certainly not learn how to think like Gaus or any of the other great mathematicians. So, what do we have to do? Well, let's have Sophia tell us about it.

[00:04:14]>> To really learn mathematics, you have to stop asking, "What trick do I use?" and start asking why am I doing what I'm doing? Why am I using this method exactly? So, it's not like there's something wrong with tricks or with following known logical steps per se.

[00:04:30]The problem is when you don't understand why you're doing them. So, what's the reasoning behind them? That's why the Gaus problem is actually something really deep. Without knowing the trick of how to do it, how would you come up with a solution?

[00:04:43]How would you rationalize it in your head? Leave your method in the comments.

[00:04:48]It's a good start to replace the sum of whole numbers from 1 to 100 with 1 + 2 + 3 plus and so on up of 2 + 100. But still, just like words, mathematical symbols are kind of language that has its limitations because our brains don't actually think in mathematical symbols.

[00:05:06]Basis, for example, shares that instead of mathematical notation, he uses squares that he stacks on top of each other. And he gives us this image as an approximation of what goes on in his head.

[00:05:18]This that you're seeing here is actually not a complete representation because we're only using 18 squares for simplicity. But imagine we had 100 squares at the base of the shape. The obvious thing about it is that it clearly looks like a triangle. And actually the number we're looking for is in the area of the triangle.

[00:05:35]Now say you immediately know what the equation for calculating the area of the triangle is. All you have to do is multiply the base time the height and divide by 2. The base is 100, the height 100. So 100 * 100 is 10,000 and 10,000 divided by 2 is 5,000.

[00:05:51]Not quite, but we're almost there. We're actually forgetting that half of the squares above the diagonal line don't count as the area for the triangle. So there are 100 little squares that only half of which is included in the triangle. So we're missing half of the area, which is 50. So now all we have to do is add it to our 5,000 and we got our answer 550.

[00:06:14]But say you didn't know the formula of the area of the triangle. Well, that's not a problem. If you notice, you can stack a copy of your triangle on top and create a rectangle. Now, we have a rectangle that is 101 squares high and 100 squares long. So, in total, there are 10,100 squares. And therefore, there are 50,50 squares in each triangle.

[00:06:37]This is basically what we have in Gaus's trick, but with triangles. Of course, in mathematics, most of the time things are not this simple and not even close. And the point is also not to get rid of your mechanical reasoning and just replace it with geometrical reasoning. That's not the point. The point is to train yourself to think in harmony with intuition and logic and also to be able to see the big picture and see why you're doing what you're doing. And of course to see the details of how you're going to achieve that. So to harmonize the big picture with the details of how you're going to get there. And again, it's not always about imagining geometric shapes. Mathematics is really diverse. If you take a look, the thing we can see is that in the end, they're like different points of view for the same mathematical reality. And it's just incredible to see this unity of mathematics.

[00:07:24]So, if you're serious about learning these, you should check out our courses on YouTube. These videos come with PDFs with more details and exercises with solutions for you to practice. You can also sign up with your email address on our website to have access to our future books and courses. link in the description.

[00:07:41]And this unity of mathematics is exactly why we were able to solve a problem in arithmetic using geometry.

[00:07:48]This is so true that actually here's a bonus question for you. How would you find the answer to the Gaus problem using probability this time? So not geometry but probability.

[00:07:59]Instead of taking the entire sequence, why don't we choose a number between 1 and 100 at random?

[00:08:05]And actually the better question is when you randomly choose a number between 1 and 100, what is the value on average?

[00:08:13]So if you didn't understand, here's an interesting example.

[00:08:17]Let's say someone came up to you with a bag full of money. There's a $1 bill, a $2 bill, a $3 bill, and so on up to 100.

[00:08:25]And let's just say that there were dollar bills for all of these integers.

[00:08:30]If you were to reach in with your eyes closed, how much money do you expect to draw? So again, if you were to choose a random number between 1 and 100, what would be its value on average? Most people would see 50 because 50 feels like the middle between 1 and 100. And if the average number is 50, then the whole sum should be about 100 * 50, which is 5,000. Here we're already extremely close. But actually the exact average is not 50 because 1 + 100 divided by 2 is actually 50.5.

[00:09:01]So the average is 50.5.

[00:09:04]Did you see what we just did there? Once you see that, the whole problem almost disappears. 100 numbers with an average value of 50.5 must add up to 5,50.

[00:09:15]So here's what Biz is trying to argue with this example. You already knew how to find the sum of whole numbers from 1 to 100, but you just didn't notice.

[00:09:24]Averages are something really intuitive for you. You don't even need to think much about them. They were taught to you since you were a kid. This is what mathematicians do with abstract concepts. They become so familiar with them that they stop feeling abstract.

[00:09:36]But that kind of familiarity takes time.

[00:09:38]You have to sit with the idea and make it feel intuitive to you. And over time, you're going to develop the necessary tools to build the rigorous foundation of it. Now, if you ever wondered, is your mind just a logical machine like a computer? And what is the best way to teach it mathematics? Well, then click on this video right here. I'll see you there.