Secret of row 10: a new visual key to ancient Pascalian puzzles | Summary and Q&A

TL;DR

Explore the surprising patterns and connections in Pascal's triangle through a colorful game involving three colors and the sum of numbers, revealing symmetrical fractals and special numbers.

Key Insights

• 👾 The game with three colors and hexagons follows algebraic rules, resulting in a colorful equilateral triangle with predictable color patterns.
• #️⃣ Certain numbers, such as 4 and numbers one more than a power of three, are special in the game, where the shortcut for determining the bottom corner color always works.
• 👾 The game has connections to Pascal's triangle, with similar addition rules and the emergence of fractal patterns.
• 👾 By exploring different color patterns and using modular arithmetic, the game reveals symmetrical fractals and self-similar patterns.

Transcript

Welcome to another Mathologer video. Today we'll take a well-earned break from all the heavy-duty algebra of the last couple of videos. Today it's all going to be super visual and super accessible, promise. Okay, to start with let me first get you hooked. Three colors: red, yellow, blue. Put down a row of ten hexagons and color them randomly. There... Read More

Questions & Answers

Q: How does the game with three colors and hexagons work?

The game involves coloring hexagons based on the sum of the colors of the hexagons above it. Red plus yellow equals blue, yellow plus blue equals red, and blue plus red equals yellow. When hexagons are added in subsequent rows, the same rules apply, resulting in an equilateral triangle composed of hexagons.

Q: Why is the bottom corner always the sum of the colors of the top corners?

The bottom corner always follows the rule of summing the colors of the top corners because the colors in the triangle are based on an algebraic calculation. When the colors of the top corners are added, a new color is determined, and this pattern continues with subsequent rows.

Q: Are there any special numbers in the game?

Yes, certain numbers are considered special in the game. For example, the number 4 is special because when a width four triangle is created, the shortcut for determining the bottom corner color always works. Additionally, numbers that are one more than a power of three (such as 10, 28, and 82) are also special.

Q: How does the game relate to Pascal's triangle?

The game is closely connected to Pascal's triangle. By coloring Pascal's triangle using the remainders of numbers divided by two or three, similar patterns and fractals emerge. The game follows similar addition rules as Pascal's triangle, leading to fascinating connections between the two.

Summary & Key Takeaways

• The video introduces a game with three colors and hexagons, where the color of a hexagon is determined by the sum of the colors of the hexagons above it.

• The game creates an equilateral triangle composed of hexagons, and the color of the bottom corner is always the sum of the colors of the top corners.

• The video explores various aspects of the game, including its relation to Pascal's triangle and the discovery of special numbers.