ZVEZDELINA: It's a game
Attributed to John Conway, a mathematician at Princeton, and it is called Conway checkers.
Conway checkers is a game which is played on an infinite board.
So it's infinite in all directions. You take a line, and you put it somewhere in the middle of your board
and you're not going to move it. And then what you do is you take a few checkers and you place them underneath the line.
Anywhere you like, it's totally random.
It's up to you what you want to do. And then you start playing by simplified checker rules.
There is only one type of move: you can jump over an adjacent checker...
and then disappear that checker.
So I can jump
vertically or I can jump horizontally.
And I can do this left, right, up, or down, so there are four possible moves
but they're all of the same type. Now
What is not allowed is if you have three checkers in a row you try to jump like this
And you land on something. Also, you cannot jump over several checkers.
That's not allowed.
You cannot jump diagonally. You're limited to jumping only horizontally, left and right, or
vertically, up and down. And of course, at some point you may jump over the line because you will be close to it.
There you go. BRADY: This is allowed?
Z: Yep, that's allowed. In fact,
Your goal is to get as high as possible over the line.
So Conway's challenge to us is if you start
below the line in whatever
configuration you wish, as many checkers [as] you wish, and you follow those simple rules,
What is the highest row above the line that you can reach?
BRADY: Oh, I'm excited! I don't know the answer!
Z: Well, what do you think? BRADY: I think the answer is going to be...
Disappointingly low.
Z: But is that what your gut feeling says? Or your experience?
BRADY: Well, part of me thinks if I can use an infinite number of checkers in any configuration
I want, I should be able to build some mechanism to push quite high, but... Z: That's right!
That's what my gut feeling said about the problem. I said, "what kind of question is this? Of course I will somehow...
create an algorithm, perhaps inductive algorithm, and reach higher and higher and higher rows!"
BRADY: And you need to just be able to keep feeding the machine.
Z: Exactly, so let's try, and let us be efficient: do I really need a whole army of checkers to reach the first row?
Well obviously if I put the checkers like this, I'm not gonna go anywhere above the line, but if I put them vertically...
everyone can play this quite easily, you can jump over and
you're done. You reached row one, alright.
Let's try row two! Now at this point, the audience should stop the video and really try it
Because it's not entirely obvious what to do.
[music]
BRADY: I think we need to put... if we put another two...
Z: Aha! BRADY: If we put another two there, we can feed it another one.
Z: So Brady has
learned a lot of mathematics and his intuition is really good. Indeed, if we start like this...
Let's see
one move
two moves
and finally three moves and we are in row two above the line.
So
We're done. Now for an extra challenge: row three
How many checkers do you think we will need?
[music]
BRADY: Two there...
I think we need another... I think we need another four.
Z: Yes, but the question is in what configuration?
BRADY: Well I think, let me see...
Z: Um, maybe you can use the smaller ones because they fit better...
BRADY: Okay, let me use the smaller ones.
So we want two there
And then we need those two there
And then we need another two there
Z: He's so good, you've seen this before!
BRADY: I haven't, no...
Z: My goodness, you're my fastest student! BRADY: How's that?
Z: Brady, you're on. Let's try...
BRADY: Oh, look, I'm doing so well!
Z: You think you can play it, or...?
BRADY: Well yeah Z: yeah, okay, alright, try.
BRADY: So, what we did before
Z: Yes...
BRADY: and then... and then that to there.
That's gone.
And that to there Z: Okay...
BRADY: That's gone, and then we need to get that to there
And that's gone. Then we need to get that to there
Z: Ahh, this looks very promising.
BRADY: Do you know what?
I don't know how to get any higher, though.
Z: So let's think. We use two,
four, and eight checkers to get to row one, two, and three. This looks...
very determined process to me. How many checkers shall we try for all four?
BRADY: You want sixteen, do you?
Z: I do, but you know what?
There isn't a configuration with sixteen.
You will need more.
The pattern breaks! I can show you the configuration
That actually wins... and there it is! So that's 20 checkers, and the viewers can try to
play with those checkers and see if you can reach the fourth row. This is actually quite a feat.
[music]
I'm not going to try, um, to try it on Brady.
This is really tough. You start with those 20 checkers, you play play play play, and you get to row four.
In fact, there are other configurations
with slightly more checkers. For example, 21.
And so the way you do it is you arrange them
on a 4 by 5. So we're going to have already 20 checkers
Which is what we had before
But in this particular situation you put one more so you end up with 21 checkers
And it's kind of a very nice symmetric situation that actually
capitalizes on our previous attempt to reach row three. So you can try this and indeed
Reach somewhere up row four. Of course, with 21 points, this game
here is less efficient than the previous one which was with 20. And you can ask yourself, "Is there a more efficient game with
19, 18, and so on?", but really that's not the point of the game. And now the challenge is on.
BRADY: Five, row five.
Z: So we have 21 here, well, how many do you think we need for all 5?
We went more than twice up from the previous case so if we go more than twice up, maybe about
50-ish?
BRADY: Yeah, 50 or 60. I think it's gonna become impossible. I'm already surprised we got to four.
I think it's gonna become impossible soon.
Z: But I don't see any restriction! What can possibly stop us from reaching the fifth row?
So there is this mysterious entity called a monovariant.
A feature that moves only in one direction. It either only increases or only decreases
and that feature prevents us somehow from reaching the fifth row.
But getting that feature,
extrapolating it from this game is actually quite complex, so let's get to work.
BRADY: Are you ready to get to work?
If so, the proof video showing that you cannot reach row five is on our second channel,
Numberphile2, which is where I put longer, more detailed videos.
This is quite a long proof, but it's really interesting, easy to follow, and it involves the golden ratio... loads.
I'll also include some links on the screen and underneath to other videos that may be of interest, including our interviews with John Conway himself.