Red-and-Black Knights NumberPhile video

[Neil Sloane]

Brady Haran (from NumberPhile) had not been here for 5 years, but he visited in March and we shot 10 videos. One of them was released today, based on an astonishing discovery by Jonas Karlsson (a member of this list) about placing red and black knights on an infinite chessboard.  There are two parts.

RED BLACK MAIN:  https://youtu.be/UiX4CFIiegM

CHESS BOARD PATTERNS: https://youtu.be/VgmDuBCayPwBest regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

[Jon Wild]

Neil, these are great, my jaw literally dropped when you pulled out the
first sheet showing the pattern at 100,000 cells. Congratulations to Jonas.

On the topic of the other video you posted recently Neil, the one you
made with Brady on awkward primes, I saw that someone called Jesper
Mikkelsen posted today an interesting update to that sequence on a
reddit thread on /r/programming. His optimized search has uncovered a
new, third plateau in the growth of the function. I find these plateaus
such strange behaviour because the growth rate of the function follows
quite a steady pattern otherwise, and I would really like someone to
figure out where it comes from. The thread is here:

https://www.reddit.com/r/programming/comments/1tb8gv6/i_built_a_world_record_exact_solver_for_the/

and Mikkelsen writes in the comments: "The longest plateau in the
Numberphile video (and for all primes up to 1024) runs 112 primes, from
N=464 to N=575 at f=69. The new run appears to have shattered that: a
plateau of 237 consecutive primes sitting at f=145, somewhere in the
N≈1100–1300 range! I will publish that whole sweep to 2048 once it's
complete."

Jon Wild


On 2026-05-12 11:19 p.m., Neil Sloane wrote:
> *Brady Haran (from NumberPhile) had not been here for 5 years, but he

> visited in March and we shot 10 videos. One of them was released today,
> based on an astonishing discovery by Jonas Karlsson (a member of this
> list) about placing red and black knights on an infinite

> chessboard.There are two parts.*
>
> *RED BLACK MAIN:https://youtu.be/UiX4CFIiegM <https://
> can01.safelinks.protection.outlook.com/?
> url=https%3A%2F%2Fyoutu.be%2FUiX4CFIiegM&data=05%7C02%7Cjon.wild%40mcgill.ca%7Cb0bb73134c7b4fb4334b08deb09e79cd%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C639142391878182856%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=p0W3nAmh%2B%2FQkAr8pBm%2BEEM5EK7hWTymNnnVH%2BGEPM58%3D&reserved=0>*
>
> *CHESS BOARD PATTERNS: https://youtu.be/VgmDuBCayPw <https://
> can01.safelinks.protection.outlook.com/?
> url=https%3A%2F%2Fyoutu.be%2FVgmDuBCayPw&data=05%7C02%7Cjon.wild%40mcgill.ca%7Cb0bb73134c7b4fb4334b08deb09e79cd%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C639142391878215761%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=zTjulsBbhIDmvX4GWUb8w%2BlWNWAfxzAVCSQ9qCY5Sfg%3D&reserved=0>*Best regards

> Neil
>
> Neil J. A. Sloane, Chairman, OEIS Foundation.
> Also Visiting Scientist, Math. Dept., Rutgers University,

> Email: njas...@gmail.com <mailto:njas...@gmail.com>
>
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[brad klee]

Harm.On.ica and I tried this as an experiment for skills 
development and produced a few variants as artifact-03: 

https://github.com/bradleyklee/Artifacts/tree/main/03-knight-quadrants

One that I like is found here: 

https://github.com/bradleyklee/Artifacts/blob/main/03-knight-quadrants/examples/brgy_hex_spiral_r120.svg

🤖 <( Is. this. a. discovery? or is. this. a mirage? )

Have fun, 

--Brad

  

[Alex Violette]

Hey Neil,

I watched both videos and I want to say, this is some pretty interesting stuff! In the second video, multiple illustrations like the one at 1:50 really stand out to me, the one at 1:50 reminds me of a fractal and I wonder if there is any fractal that resembles its shape.
It's also got me thinking about these questions:
1.) What is the density of the colors throughout the entire grid? I also noticed with the knight problem that as more and more colored knights got added, the grid became more and more white, what is the density of the white space as the number of knights goes up(Provided it is positive)?

2.) Lets categorize the primes into categories for both all colored pieces and spots where no pieces occupy them. Are there any configurations where the prime distribution is "biased" at all?

3.) Is it possible to make a setup where the grid never "stabilizes" aka it never after a certain point settles on a final pattern?
Speaking of "astonishing discovery," I have some stuff that is also insane but I admit it's not that ready for public viewing yet(doesn't help that I am not that experienced with coding for that stuff though a test run I did sometime ago for a subset of it worked fine). I am still grinding sigma stuff out as we speak for right now via code running. Also, I recall watching the video you were in about Dungeon Numbers back in 2020 when I was in college. That was fun!

Best,

Alex Violette

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