The Brussels Choice - Numberphile

TL;DR

A mathematical game is explored where players try to reach the number 1 by either doubling or halving the digits of a given number.

Transcript

So Brady, my good friend Éric Angelini, who lives in Brussels, he came up with the following operation. This is an operation, you can take a number and you can transform it. If you start off with 123, if you look at the first two digits, it says 12. You're allowed to half that. Divide it by 2 and you would get 63. That's a legal operation, take a s... Read More

Key Insights

• 👾 The game involves manipulating digits by either halving or doubling them.
• 👾 Reaching 1 is the goal of the game, and the operations are reversible.
• ❤️‍🩹 Numbers ending in 0 or 5 cannot be reduced to 1.
• #️⃣ The number of steps needed to reach 1 from a given number is at most 12 times the number of digits in the original number.
• 🎮 The game can be played by choosing different sub-numbers to manipulate.
• ❤️‍🩹 The game has two connected components: numbers ending in 0 or 5 and numbers not ending in 0 or 5.
• #️⃣ The greedy algorithm can be used to find the biggest number that can be reached from 1 in a certain number of steps.

Questions & Answers

Q: How does the game work?

The game involves starting with a number and either dividing a sub-number by 2 if it's even or doubling it. The goal is to reach the number 1.

Q: Can you give an example of how the game is played?

For example, starting with the number 123, you can either halve the first two digits to get 63, double the 2 to get 143, or double the 3 to get 126. The choices of which digits to manipulate are arbitrary.

Q: Is it always possible to reach 1 in the game?

Yes, it is always possible to reach 1 if the original number does not end in 0 or 5. However, numbers ending in 0 or 5 cannot be reduced to 1.

Q: How many steps does it take to reach 1 from a given number?

The number of steps to reach 1 varies depending on the number, but a lower bound is 12 times the number of digits in the original number.

Summary & Key Takeaways

• In the game, players start with a number and can either divide a sub-number by 2 if it's even or double it.

• The goal of the game is to reach the number 1 using these operations.

• The operations are reversible, meaning if you can go from x to y, you can also go from y to x.