Red & Black Knights (extraordinary result) - Numberphile

Added: 2026-05-19

[00:00:00]Remember one of the first videos we did together was about a knight that gets trapped and we can't repeat square. We have to always go to a new square. The night is moves around on a square spiral in an 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.

[00:00:31]>> Lowest.

[00:00:32]>> 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 two and and so on. And he keeps going. From two, he can go to five. From five, he can go to eight.

[00:00:53]>> He can't go back to one he's been to.

[00:00:55]>> That's right. It's got to be a new square, an unoccupied square. From eight, he can go to three, and from three, he can go to six, and from six, he can go back to So, he's filled in one, and then he fills in four, and he keeps going. So, so it looks like he he's going to go on, but in 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 eight things he could move to, he's visited already. So, it's a finite sequence. The sequence of squares that he stepped on. Today's video we have a lot of knights, not just one. And we put them down in a way they are courteous knights, very friendly, gentlemanly knights, and they don't want to attack each other. They want to keep their distance from all the other knights. And we we just put them down.

[00:01:40]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 knight, if it's not being attacked by any existing knight, you put a knight there. So that we put a knight at zero.

[00:01:56]That's a knight on zero. And we will go around the spiral. Now we come to square one. Now square one is not attacked by any can't see any knights from it. This knight is not a knight's move from that.

[00:02:06]We So we can put a knight there >> because the first knight couldn't catch him. The first knight couldn't catch him. It's not in the domain of any existing knight. Right? Now we come to two. Two isn't two is also unattacked and three is unattacked. Four would be attacked by this knight on square one.

[00:02:24]So we can't put a knight on four. So we we can't put knights at 12 because of that or 13 14 15. No. The first one we come to. No. 16 is is attacked by that.

[00:02:36]17 18 19 20 is the first time we can put down another knight. All right, let me keep going. 25 is free. So, we put a knight at 25. 26. No, no, no, no, no. 30 is good. 31, 32, 32, they're all attacked. 35.

[00:02:53]>> Well, it's really interesting because you start thinking, oh, hang on now.

[00:02:56]That one's going to catch that one. And that like they all start coming into play in different ways.

[00:03:01]>> Yeah, it's complicated. We get some interesting patterns which you you might think would be random or they you might think they'd be regular. Well, you'll see what happens. 40 is okay. Nobody is attacking 40 or 41 or 42.

[00:03:14]49. They're all good. Woohoo.

[00:03:18]53 is out. Yes. And 54 is okay. And 65 is 64 is not, but 65 is free. So, we got a cluster of five. We can't do 67, 68, 69, 70 we can do. And we've filled up a cluster of five and so on. And it keeps going. We got a cluster of four in the middle. And then we get clusters of five around it. And that pattern sort of continues.

[00:03:54]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 a very precise mathematical sense. In this quadrant, we have clusters of five separated by single knights. Upon this vertical line, we have clusters of 2 4 2 4 2 4 that keep going forever. And similarly in all the others. So, so this is a periodic structure. Pretty but regular. So, it's pretty nice. But if we have knights of two colors, something totally different and unexpected and amazing. Red and black knights. All right. And they take turns. And the rule is when it's black's turn to place a knight, he places a black knight at the first square that isn't attacked by a red knight. They don't mind collaborating.

[00:04:49]Black can be knight moves from black knights but not from red knights.

[00:04:53]>> So if I'm on a spot and I'm a black knight, it's okay if a black knight could catch me because he won't catch me.

[00:04:59]>> He won't. They were we're friends. We got two armies really. They got the black knights and the red knights and they're they're really competing for Europe. You have to think of the great plane of Europe and um these are the knights.

[00:05:11]>> Neil, who even thought of this doing this? Jonas Carlson in Sweden sent me a letter with describing some some interesting problems that he'd been looking at and this was one of them. And he said, "This this blows my mind. This is so incredible." He started off, as I just did with you, with black knights, and we get a periodic structure. No mystery at all. But wait till you see what happens with the red knights. Oh.

[00:05:36]Oh, boy. So the rule is you go around the spiral again and you place a black knight on the first square that is not occupied and is not attacked by a red knight. Red gets to play. They take turns. Red puts down a a red knight and the first square. It can be attacked by a red knight, but it must be unoccupied and not be attacked by a black knight.

[00:06:02]Reds and blacks are very respectful of each other. Black goes first and puts his knight at the lowest square, not occupied. Okay, red goes next. One is free. It's not attacked by that black knight. So, it goes there. Black's turn to play. Black claims that square three is not attacked by a black knight.

[00:06:24]Black's turn to play. Black would ideally like to place a knight there, but that square is under attack by that red knight. So, it can't go there. So it goes to the next one that it can. Now it's red's turn to play. This square was attacked by this red knight, but that is fine. The red knights collaborate, cooperate with each other.

[00:06:45]>> So you can fill squares that have come before if your opponent hasn't filled it. Yes, obviously.

[00:06:50]>> But some squares are going to be unoccupied at the end of the day.

[00:06:54]All right, black's turn. Black can't go there because of that red knight or there, but it can go there. Red's turn.

[00:07:03]Red plays here. Black's turn. Black it can't go here because it would be attacked by that red knight. It's got to avoid any square that's being attacked by a red knight. And we It's got to be the smallest unoccupied number. So, I think it could go there. There's no red knight within a knight's move of that square. All right, it's red's turn. Red continues from where it left off. Seven.

[00:07:30]Uh, seven is no good because of that.

[00:07:32]Eight is no good because of that black knight. 10. 10 is free for red. So now it's black's turn to play. Can't go there because of that. It can't go there.

[00:07:45]It can No, it can't. It can go there.

[00:07:49]All right. Black just played on square 15. Red's turn. Red can go here. Black.

[00:07:56]We left black at 15. And we can put a black here. Okay, red. Back to red.

[00:08:04]Here's 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 24 can be red. And it keeps going like that forever. This is not a game that ends.

[00:08:19]And it's not really a game. The game, you could say it's a matter of seeing um who can control the most territory. But for the moment, we're just putting down knights and looking at the pattern. And >> I am. Yeah, you got me wondering that.

[00:08:32]Yeah, obviously it's all predetermined, but did playing first give you an advantage to have more territory or Yeah. No, >> in the end, no. Didn't matter. It evens out. It It's pretty obvious it's going to even out in this kind of game.

[00:08:47]There's no particular advantage. And what you get, and this Jonas sent me a picture of this. He 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.

[00:09:11]This bottom quadrant here is all red and it looks there's an awful lot of black here and here it's 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 100,000 cells.

[00:09:35]And you can see this is a a continuation of the first thousand cells.

[00:09:44]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 knights. And then there's a strip of black knights. And here you've got mixed.

[00:09:59]>> So this little red corner you showed me is the start of this section here.

[00:10:04]>> But it's only a strip.

[00:10:06]>> Yeah.

[00:10:06]>> And it's amazing that you would get this. Where do these strips come from?

[00:10:12]You'd think it would be totally random.

[00:10:14]>> Yeah. You had all this black. You can still see the bits of white in there, too, though. These are the blank squares.

[00:10:18]>> The blank squares. Yes. Yeah.

[00:10:20]>> Yeah. How do these islands form? What?

[00:10:22]>> Yes. Yes. It's mysterious.

[00:10:24]>> Yeah. It's fantastic. Did Did you catch that? All right. Let me show you the next stage in the evolution. That was that was 100,000 cells. When we get up to a million cells, it's fantastic.

[00:10:41]Michael Branicki got involved at this point and he he contributed some drawings. This is one of his.

[00:10:47]>> There are my islands. They're like, >> "Yeah, they're the islands." And there are islands. There they are. And they stay there. I mean, this this is a subset of this. It gets it's the same picture, but just bigger. This is a million squares. As we go up, the red the black expands and gets more and more black. By the time we get to 64 million, 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 in terms of uh Stendal's novel where the red represents the military and the black represents the church whether they're going to be join 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.

[00:11:44]>> Does this black and red now last forever or >> Yes. Yes. It just gets bigger and bigger.

[00:11:48]>> Yeah.

[00:11:50]>> Yeah. They grow and the strips are thin strips and people who are undecided whether they're going to be red or black. And I don't know if this has any political implications, but it certainly is astonishing.

[00:12:08]>> It's very unexpected that it for so long it was making patterns and then when it suddenly just said, "No, that's it.

[00:12:15]>> That's it.

[00:12:15]>> We're done."

[00:12:16]>> Yeah. It's contagious. You could say >> amazing.

[00:12:22]>> Amazing. Yeah.

[00:12:33]Do you know what? Has anyone done it with three colors now?

[00:12:36]>> That would be interesting, wouldn't it?

[00:12:37]>> Yeah.

[00:12:38]>> Yeah.

[00:12:39]>> Let's do it. Let's do it now.

[00:12:42]>> Should we do it?

[00:12:43]>> No. Because it'll take a long time to get enough colors for it. It's going obviously going to be unsettled until the jelly sets.

[00:12:50]>> Will it settle? Will three colors settle? Well, how will three colors settle? because you haven't got those quad those obvious quadrants like you do with a square board. Like what will three colors do?

[00:13:01]>> I'll investigate.

[00:13:03]Giving you credit for suggesting it unless you want to work on it.

[00:13:06]>> I I don't think I have the firepower.

[00:13:08]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.

[00:13:28]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 it d the sequence dies at that