Introducing "Beautiful Ideas"

Puzzles, brain teasers, paradoxes, and deep thoughts

One of the things that first drew me to the study of mathematics and economics is the feeling of pleasure and awe I experienced while following a rigorous line of reasoning out to its conclusion. The unexpected twists and turns of an ingenious argument, the elegance of a mathematical proof, the symmetry of an equation: when understood, these are things of true beauty, no different in their effect on the mind and soul than a Beethoven string quartet or a Caravaggio painting.

Beautiful ideas bring me joy, and I want to share some of that joy with all of you here. So from time to time, I’m going to post logic puzzles, brain teasers, paradoxes, and mathematical and economic concepts that I find beautiful or wondrous in some way. The TGS team and I will also sometimes use these posts as an opportunity to experiment with new audio-visual formats.

The first “Beautiful Ideas” I have for you take the form of brain teasers. Below I’m posting videos that describe two different games. Your task is to figure out a strategy for each that will guarantee that you win the games. The first video is available to everyone, but the second one is available only to paying subscribers. When you devise your winning strategies, post them in the comments. In a couple days, I’ll release videos with solutions to the puzzles.

I’ve written the puzzles out, but if you want the full experience, you’ve got to watch the accompanying videos, which feature wonderful animations by Nikita Petrov illustrating the problems.

I’m looking forward to seeing what you come up with!

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PUZZLE #1

There’s a box. It contains 20 balls. Two players compete with one another by alternately removing either one, two or three balls from the box. They start with 20 balls and they take turns. Each must take at least one but no more than three balls from the box. The player who takes the last ball loses the game.

Now, imagine that you are one of these players, and that you have the first move. What would you do? If you want to win the game, how many balls do you take to be sure that you can win? And I want you to explain your reasoning.