Chapter 1: The Puzzle Begins

On the 21st of August, I found it fitting to pose an intriguing question: What is the 821st number? The setup involves two players taking turns counting, starting from 1 and adding one more to the last number mentioned by the opponent.

As a hint, consider summing numbers from 1 to n! Take a moment to pause here, grab some pen and paper, and try to solve it on your own. Once you're ready, continue reading for the solution!

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Section 1.1: Understanding the Counting Sequence

When I state one number, you then respond with two. This alternates, so it goes 1, then 1, 2, followed by 1, 2, 3, and so on. Each time a player concludes with number n, we collectively have stated a total of 1 + … + n numbers!

Thanks to Gauss' brilliant insight, we can derive our answer.

Do you know how to derive this result? It requires some experimentation. Grab a calculator and try out different values for n. For instance, if you choose n = 40, you'll find...

This result is just one less than 821! Let's analyze what's happening here. When n equals 40, it indicates that we have counted from 1 up to 40, alternating turns.

Section 1.2: Arriving at the Solution

Because 821 - 820 equals 1, the next player simply has to say one number!

And there you have it: that is our answer.

Photo by yamasa-n on Unsplash

Isn't that fascinating? I'm curious about your thought process on this challenge. Please share your insights in the comments below!

Chapter 2: Discover More Math Puzzles

...And don’t forget to check out these exciting math puzzles on Medium!

Math Puzzles

The best math puzzles on Medium

Algebra, Geometry, Calculus, Number Theory, and more.

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