Gauss Sum Formula: How a 7-Year-Old Instantly Added 1 to 100

The Boy Who Surprised His Teacher

At the age of 7, Carl Gauss began his studies at St. Catherine’s elementary school. His teacher was the school’s headmaster, a man named Büttner. Gauss quickly astonished him with his mathematical abilities — a story that many of you have probably heard.

One day, Büttner wanted to get on with his own affairs and decided he needed something to keep his students busy and out of his way. What better distraction than a difficult math problem? He asked the class to find the sum of all numbers from 1 to 100. The students got to work, adding each number one by one. Büttner’s frustration can only be imagined when, just a few minutes later, little Gauss announced the answer: 5050.

How the Gauss Sum Formula Works

How did he get there so fast? Add 1 and 100 — you get 101. Add 2 and 99 — again 101. Add 3 and 98 — 101 once more. And so on, all the way to the last pair: 50 and 51. The result: instead of laboriously adding 100 numbers in sequence, you simply add 101 exactly 50 times — or multiply 50 by 101. That gives you 5050. Simple and elegant.

Of course, this story is an inspiring legend, and no one knows for certain which method Carl actually used. But the principle itself — grouping terms into pairs — reveals the beauty of mathematics. To find a solution, it helps to look at the problem from a new angle rather than doing what everyone else does.

From a Classroom Trick to a Universal Formula

This insight leads directly to the general formula for the Gauss arithmetic series:

1 + 2 + … + n = n(n+1)/2

But this is only the beginning.

A Challenge for You

Here is a question to think about: what is the sum of squares 1² + 2² + 3² + … + n²?

Share your answer in the comments and explain how you arrived at it. Think, explore, and enjoy the beauty of mathematics!