Alpha-Evolve

5/17/2025

A Easy Reading guide to the kissing-number problem—how many spheres can touch one central sphere—plus a look at how Google DeepMind’s AlphaEvolve is advancing the frontiers of this classic puzzle.

Kissing Number Made Easy

(and How AlphaEvolve Keeps Breaking New Ground)

1. What Is the Kissing Number?

Picture a gentle bump—like two billiard balls kissing.
The kissing number answers one simple question:

How many identical balls (or coins) can all touch a single central ball without overlapping each other?

Think of it as counting the maximum number of “hugs” a sphere can get at once.

2. Simple Cases You Can See

| Dimension | What you’re really packing | Maximum “kisses” | Easy picture |
|-----------|---------------------------|------------------|--------------|
| **1-D**   | Points on a line          | **2**            | One on the left, one on the right |
| **2-D**   | Coins on a table          | **6**            | Six coins form a daisy around one coin |
| **3-D**   | Marbles or oranges        | **12**           | Twelve oranges wrap one orange—no room for a 13th |
| **4-D**   | Hyperspheres              | **24**           | Needs math imagination! |

Kid-Friendly Walk-Through

  • 1-D: Slide one ant to each side of a breadcrumb—only two can touch.
  • 2-D: Draw a coin, then slide six more so each coin “hugs” the middle like flower petals.
  • 3-D: Stack oranges: three at the bottom, then layers until you reach 12 around one. Try adding a 13th—no gap is left!

3. Why Should Anyone Care?

  • Atoms & Crystals – Metals arrange atoms in patterns tied to kissing numbers, affecting hardness and conductivity.
  • Error-Correcting Codes – The same geometry protects your mobile-phone texts from noise.
  • Pure Puzzle Fun – From classrooms to puzzle YouTube, the problem delights everyone.

4. Proven Kissing Numbers (So Far)

| Dimension (*n*) | Proven value *k(n)* |
|-----------------|---------------------|
| 1 | 2 |
| 2 | 6 |
| 3 | 12 |
| 4 | 24 |
| 8 | 240 |
| 24 | 196 560 |

Other dimensions remain mysteries—perfect challenges for modern AI.

5. AlphaEvolve Joins the Adventure

AlphaEvolve is Google DeepMind’s new science-AI platform that evolves code and proofs much like natural selection for ideas. In early 2025, AlphaEvolve nudged the known lower bound for the kissing number in 11 dimensions from 592 to 593—a brand-new record!

Beyond geometry, AlphaEvolve has:

  • Improved long-standing matrix-multiplication algorithms.
  • Helped cut data-center energy use.
  • Generated thousands of candidate proofs, selecting the strongest automatically.

Its ability to explore vast idea-spaces makes AlphaEvolve a perfect partner for cracking tough puzzles like the kissing-number problem.

6. Try It at Home

  1. Coin Flower – Tape one coin, trace it, then arrange six more around it. Color them as petals.
  2. Orange Challenge – Use 13 identical marbles: prove the 13th can’t touch the middle one.
  3. Mini-AlphaEvolve – Code a small program that searches for the best circle packings on-screen; watch “population” ideas evolve.

7. Key Takeaways

  • The kissing number measures how many friends a sphere can hug.
  • Answers are easy in 1-D, 2-D, and 3-D—but skyrocket in complexity beyond.
  • AlphaEvolve is already pushing the frontiers, hinting at a future where human curiosity teams with machine creativity to unlock even higher-dimensional secrets.

Stay Curious! Keep an eye on AlphaEvolve updates. Each breakthrough might inch us closer to solving higher-dimensional kissing-number mysteries.

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