---
title: 'Red & Black Knights (extraordinary result) - Numberphile'
source: 'https://youtube.com/watch?v=UiX4CFIiegM'
video_id: 'UiX4CFIiegM'
date: 2026-06-30
duration_sec: 820
---

# Red & Black Knights (extraordinary result) - Numberphile

> Source: [Red & Black Knights (extraordinary result) - Numberphile](https://youtube.com/watch?v=UiX4CFIiegM)

## Summary



## Transcript

Remember one of the first videos we did together was about a knight that gets trapped. And we can't repeat the square. We have to always go to the new square. The knight moves around on a square spiral.
In the infinite chessboard and we've numbered the squares. And you start at zero, so I'll put a penny where you've been.
And where you move to has to be the lowest square that you haven't visited. Lowest. The lowest, the smallest number you haven't been to. So from zero, we could go to 9, 11, blah, blah, blah.
So he goes here. Okay, now where does he go? I put a penny to mark where he started. He can go to 2. And so on. And he keeps going from 2, he can go to 5, from 5, he can go to 8.
He can't go back to 1, he's been to. That's right. It's got to be a new square, an unoccupied square. From 8, he can go to 3, and from 3, he can go to 6, and from 6, he can go back. So he's filled in 1, and then he fills in 4, and he keeps going.
So it looks like he's going to go on, but in fact, after a while he runs out of squares to go to. He gets trapped. He ends up at a square where all of the 8 things he could move to, he's visited already.
So it's a finite sequence, the sequence of squares that he stepped on. In today's video, we have a lot of nights, not just one. And we put them down, in a way, they are courteous nights, very friendly, gentlemanly nights.
And they don't want to attack each other. They want to keep their distance from all the other nights. And we just put them down. The rule is, you walk along the spiral. And when you come to a square, if it's not in the domain of any existing night,
if it's not being attacked by any existing night, you put a night there. So we put a night at 0. That's a night on 0, and we will go around the spiral. Now we come to square 1. Square 1 is not attacked by any, can't see any nights from it.
This night is not a night's move from it. So we can put a night there. Because the first night couldn't catch him. The first night couldn't catch him. It's not in the domain of any existing night. Now we come to 2. 2 isn't. 2 is also an attack, and 3 is an attack.
4 would be attacked by this night on square 1. So we can't put a night on 4. So we can't put nights at 12, because of that, or 13, 14, 15, no. The first one we come to, no, 16 is attacked by that. 17, 18, 19, 20 is the first time we can put down another night.
Then we keep going. 25 is free. So we put a night at 25. 26, no, no, no, no. So 30 is good. 31, 32, 32. They're all attacked. 35. It's really interesting, because you start thinking, oh, hang on now, that one's going to catch that one.
They all start coming into play in different ways. Yeah, it's complicated. We get some interesting patterns, which you might think would be random. Or they might think they'd be regular. Well, you'll see what happens. 40 is okay. Nobody is attacking 40, or 41, or 42, 49. They're all good.
Oh, 53 is out. Yes. And 54 is okay. And 65 is, 64 is not. But 65 is free.
So we've got a cluster of 5. We can't do 67, 68, 69, 70 we can do. And we've filled up a cluster of 5 and so on. And it keeps going. We've got a cluster of 4 in the middle. And then we get clusters of 5 around it.
And that pattern sort of continues. And here's a picture of what it looks like after about a thousand steps of the spiral.
And you can see it's periodic in a very precise mathematical sense. In this quadrant, we have clusters of 5 separated by single nights.
Up on this vertical line, we have clusters of 2, 4, 2, 4, 2, 4 that keep going forever. And similarly in all the others. So this is a periodic structure. Pretty but regular.
So it's pretty nice. But if we have nights of two colours, something totally different and unexpected and amazing. Red and black nights. All right. And they take turns. And the rule is when it's black's turned to place a night.
He places a black night at the first square that isn't attacked by a red night. They don't mind collaborating. Black can be night moves from black nights, but not from red nights.
So if I'm on a spot and I'm a black night, it's okay if a black night could catch me. Because he won't catch me. We're friends. We've got two armies really. They've got the black nights and the red nights. And they're really competing for Europe.
You have to think of the great plane of Europe. And these are the nights. Neil, who even thought of this doing this? Jonas Carlson in Sweden sent me a letter with describing some interesting problems that he'd been looking at.
And this was one of them. And he said, this blows my mind. This is so incredible. He started off as I just did with you with black nights and we get a periodic structure. No mystery at all.
But wait and you see what happens with the red nights. Oh boy. So the rule is you go around the spiral again and you play some black night on the first square that is not occupied and is not attacked by a red night.
Red gets to play. They take turns. Red puts down a red night at the first square. It can be attacked by a red night, but it must be unoccupied and not be attacked by a black night.
Reds and blacks are very respectful of each other. Black goes first and puts his night at the lowest square not occupied. Okay. Red goes next. One is free. It's not attacked by that black night. So it goes there.
Blacks turn to play. Black claims that square. Three is not attacked by a black night. Blacks turn to play. Black would ideally like to play tonight there, but that square is under attack by that red night.
So it can't go there. So it goes to the next one that it can. Now it's reds turned to play. This square was attacked by this red night, but that is fine. The red nights collaborate, cooperate with each other.
So you can fill squares that have come before if your opponent hasn't filled it. Yes, obviously. But some squares are going to be unoccupied at the end of the day. All right. Blacks turn. Black can't go there because of that red night or there, but it can go there.
Reds turn. Red plays here. Blacks turn. Black can't go here because it would be attacked by that red night. It's got to avoid any square that's being attacked by a red night, and it's got to be the smallest unoccupied number.
So I think it could go there. There's no red night within a night's move of that square. It's reds turned. Red continues from where it left off. Seven is no good because of that.
Eight is no good because of that black night. Ten. Ten is free for red. So now it's black's turn to play. It can't go there because of that. It can't go there.
It can go there. All right. Black displayed on square 15. Reds turn. Red can go here. Black. We left black at 15, and we can put a black here. Okay. Red. Back to red. His red. The last red was at 12.
We can't do 13. We can't do 14. 15 is occupied. We can do 24, I think. 20 thought can be red. And it keeps going like that forever. This is not a game that ends, and it's not really a game.
The game, you could say, it's a matter of seeing who can control the most territory. But for the moment, we're just putting down nights and looking at the pattern. I am. You got me wondering that, yeah. Obviously, it's all predetermined.
It's predetermined. But did it play in first, give you an advantage to have more territory? In the end, no. It didn't matter. It evens out. It's pretty obvious it's going to even out in this kind of game.
There's no particular advantage. And what you get, and this Jonas sent me a picture of this. He showed me what you get after a thousand squares. I've indicated the starting square with a little black circle.
And they alternate black. It's this board, but done for a thousand squares. And you can see some strange things happened. This bottom quadrant here is all red.
And it looks, there's an awful lot of black here. And here is all mixed. So it's not clear what's going to happen. That's a thousand, a thousand squares. And Jonas said it is totally unbelievable what happens. This is for a hundred thousand cells.
And you can see this is a continuation of the first thousand cells. This picture is a subset of this. It's embedded. It's in the middle.
You can see this chunk of red here has become a wide strip of reds. In this strip there are only red nights. And then there's a strip of black nights. And here you've got mixed. So this little red corner you showed me is the start of this set.
Yes, yes. But it's only a strip. And it's amazing that you get this. Where do these strips come from? You'd think it would be totally random. You can still see the bits of white in there too though. These are the blank squares.
Where's the blank squares? Yes. How do these islands form? Yes. Yes. It's mysterious. It's fantastic. Did you catch that? Let me show you the next stage in the evolution.
That was a hundred thousand cells. When we get up to a million cells, it's fantastic. Michael Brandicke got involved at this point.
And he contributed some drawings. This is one of his. There are my islands. They're like, yeah, they're the islands. And there are islands. There they are. They stay there. This is a subset of this. It's the same picture, but just bigger.
This is a million squares. As we go up, the black expands and gets more and more black. By the time we get to 64 million, a black has totally taken over one quadrant of the board.
In fact, two quadrants separated by a strip, which are kind of indecisive. They can't really decide whether they're Republicans or Democrats or in terms of a standar's novel
where the red represents the military and the black represents the church. Whether they're going to be joined the military or join the church. Solid black in two quadrants, two quarters of the whole infinite board,
or solid red in the top half. Does this black and red now last forever? Yes. It just gets bigger and bigger. Yeah. They grow. And the strips are thin strips and people who are undecided
whether they can be red or black. I don't know if this has any political implications, but it certainly is astonishing. It's very unexpected that for so long it was making patterns and then when it suddenly just said no, that's it.
That's it. We're done. It's contagious. Amazing. Amazing.
Do you know what? Has anyone done it with three colours now? That will be interesting, wouldn't it? Yeah. Let's do it. Let's do it here. Should we do it? No, because it'll take a long time to get enough colours for it.
It's obviously going to be unsettled until the jelly sets. Will it settle? Will three colours settle? Well, how will three colours settle? Because you haven't got those quadrants like you do with a square board.
But what will three colours do? I'll investigate. Giving you credit for suggesting it unless you want to work on it. I don't think I have the firepower. We'll show you what happens with three pieces in a bonus video over on Number File 2.
We'll also show you what happens with more pieces and pieces which move in different ways. Trust me, it gets pretty crazy. Check out the links below. It gets stuck, it cannot move. Every square, every one of the eight squares around it,
that it could move to, it's already been to. So the sequence dies at that point.