TL;DR
In any coloring of a 4×4×4 cube with three colors, you can always find 12 cubes of the same color that form 6 pairs with equal distances between their centers.
Key Takeaways
- 1The puzzle requires a 4×4×4 block of 64 sugar cubes, each colored white, blue, or brown
- 2You must prove 12 cubes of identical color exist that can be divided into 6 pairs with equal center-to-center distances
- 3The solution must work for any possible color assignment across all 64 cubes
- 4This is a combinatorial geometry problem from the 2024 British Math Olympiad
Chapters
1. Puzzle Setup
A 4×4×4 block contains 64 sugar cubes, each assigned one of three colors: white, blue, or brown.
2. The Challenge
Find 6 pairs of same-colored cubes (12 cubes total) where the distance between the centers of each pair is equal across all 6 pairs.
3. Proof Requirement
Demonstrate that such a configuration exists for any possible color allocation, making this a universal proof rather than a single example.
Key People & Entities
- British Math Olympiad
- Source of the 2024 puzzle
Glossary
- 4×4×4 cube
- A three-dimensional arrangement of 64 unit cubes stacked in a 4-by-4-by-4 formation
- Combinatorial geometry
- Mathematical field studying geometric structures through counting and discrete methods
- Pigeonhole principle
- Mathematical concept stating that if n items are placed into m containers where n > m, at least one container must hold more than one item