The 64 sugar cubes puzzle

3Blue1Brown · 0m · transcribed Jul 2026

Section Insights

# 0:00

Introduction to the Puzzle

What is the new puzzle of the month?

The puzzle involves 64 sugar cubes arranged in a 4x4x4 block, sourced from the 2024 British Math Olympiad.

The puzzle is based on a mathematical problem.
It involves a 4x4x4 arrangement of sugar cubes.
The challenge is derived from a prestigious math competition.

# 0:09

Color Pairing Challenge

What is the challenge regarding the sugar cubes?

The challenge is to prove that you can find 6 pairs of cubes, all of the same color, from the 64 cubes.

Cubes can be white, blue, or brown.
The goal is to find 12 cubes that can be paired by color.
The solution requires combinatorial reasoning.

# 0:19

Equal Distances Requirement

What additional condition must be met for the pairs of cubes?

The lengths of the lines connecting the centers of the cubes in each pair must all be the same.

The challenge involves geometric considerations.
All pairs must have equal distances between their centers.
This adds complexity to the color pairing task.

# 0:29

Universality of the Solution

Is the solution dependent on the color allocation of the cubes?

No, the solution must hold regardless of how the colors are allocated among the 64 cubes.

The problem is universally applicable to any color arrangement.
It emphasizes the robustness of the mathematical proof required.
The challenge is to demonstrate this universality.

# 0:39

Conclusion of the Challenge

What is the final assertion about finding the pairs?

You can always find 12 cubes of the same color that can be divided into 6 pairs with a common distance.

The conclusion reinforces the existence of a solution.
It highlights the importance of combinatorial and geometric reasoning.
The challenge is both a mathematical proof and a puzzle.

Transcript

0:00 It's time for a new puzzle of the month. This one comes from a problem on the 2024 British Math Olympiad. The setup is to have 64 sugar cubes arranged in a 4x4x4 block. Each cube is either white, blue, or brown. Your challenge is to prove that you can find 6 pairs of cubes, such that all 12 individual cubes involved are the same color, and such that the lengths of these six lines, the lines connecting the centers of the cubes in a given pair, are all the same.

0:28 And that's 6 equal distances. So just to be clear, you have to show that no matter what the allocation of colors to these 64 different spots in our big 4x4x4 cube is, you can always do this. You can always find 12 little cubes of the same color that can be divided into 6 different pairs such that there's a common distance between the centers of each pair.

Summary

In this puzzle, the goal is to demonstrate that within a 4x4x4 cube of 64 sugar cubes, each colored either white, blue, or brown, it is always possible to find 12 cubes of the same color that can be grouped into 6 pairs with equal distances between their centers. This is a combinatorial problem that leverages the principles of symmetry and the pigeonhole principle.

- The cube consists of 64 sugar cubes arranged in a 4x4x4 formation.
- Each cube can be one of three colors: white, blue, or brown.
- The challenge is to find 12 cubes of the same color that can be paired into 6 pairs.
- Each pair must have the same distance between the centers of the cubes.
- The solution relies on the symmetry of the cube and the limited number of colors.
- By the pigeonhole principle, with 12 cubes of the same color, there are enough combinations to form pairs.
- The distances between pairs can be calculated based on the coordinates of the cubes in the 3D space.
- The problem guarantees that no matter how the colors are distributed, such pairs can always be found.