Okay, I wanted to tell you about the number 2,084.
It arises at the end of a little story.
This particular one came about on a chessboard.
This is a really big chessboard, this is an infinite board.
And so I'm going to number the squares. So I'll start off, I'll pick any square,
and number it 1. And now we work in a spiral around it.
And we number them. It's called a square spiral. And we go around and around and around until we've filled up the whole board.
Now what we do is we take a knight, an ordinary chess knight, and we start him off at square number one.
And now we look at all the squares that he can jump to. The knight could move from 1 to 12.
Or from 1 to 18. The rule is we always go to the smallest square and we can't repeat a square,
we have to always go to a new square. So from 1 it looks to me like we could get to 10 - there are 8 squares and 10 is the smallest.
So we're going to get a sequence which is the sequence of the squares where the knight goes to.
So it starts off at 1, its first move is to the smallest number that it can go to - that it hasn't already been to.
So it can go to 10. From 10, aha! It can move back to 3.
So 10 goes to 3. So the third term in the sequence is 3.
Then from 3 we could go to 6. 6 is obviously the next number.
And then 9, 4 - so that's the sequence.
We do that forever. It would be nice if we could do it forever.
But something strange happens. After we've been going on for a long time, it gets trapped.
In fact, after 2,016 steps it gets stuck, it cannot move.
Every square, every one of the 8 squares around it that it could move to, it's already been to.
So it di- the sequence dies at that point. After 2,016 steps and the last number that it reaches is 2,084.
And I have in my notebook here a picture of the spiral. So you can see it starts off here,
and the colours, it gets lighter and lighter. It starts off fairly dark and then we have blue and green and yellow.
And then it moves around, there are very strange long lines of points where it moves in a straight line, this knight.
Moving 2 and 1, 2 and 1 and so on. But eventually it gets to this red dot.
And there it gets stuck. All of the 8 squares it could possibly move to it's already visited.
(Brady: It's deliciously arbitrary isn't it? And unexpected that it could - you would have that some pattern or system had emerged where)
("oh, it's gonna go forever now." Does that surprise you?)
It's wonderful! Yeah, it surprises me! I think it's wonderful! It's uh...yeah, it's a strange sequence.
The missing numbers - that's another sequence. So if we look at the ones it doesn't visit there are infinitely many, that's an infinite sequence.
And the smallest one it doesn't get to is 961.
You can do it with other chess pieces. You can do it with a castle, a rook. You could do it on a chess board of a different shape,
you could do it on the kind of chess board that you get if you just - just one quadrant. One quarter of the infinite board.
So let's pretend we have our chess board and we number the squares just as we did in the spiral.
Here we're going to number them in - along diagonals. More precisely along antidiagonals.
2, 3 and so on.
(Brady: Then what, same game again?)
You play the same game.
1, and what's the smallest square it can move to? Well it can only move to 8 or 9 so it has to go the smaller of them.
So the sequence goes 1, 8. Now from 8 we could move to some big numbers here,
but the smallest number is obviously 6.
So 1, 8, 6. And then to 2. And then to 12 it looks like? And so on.
So 1, 8, 6, 2, 12. Again, the sequence is finite. At step 2,402 it reaches 1378 and there are no squares it can move to.
It dies at that point. Again it's colour coded. It starts off in the corner and as the trajectory moves along the colours get lighter and lighter,
and you see that red X there? That's where it gets trapped.
(Brady: You love it don't you?) - I do.
So you can do the same thing with the rook. With a queen it doesn't get stuck, it keeps going.
(Brady: Do you play chess?) - No. Not any more. I retired. I retired at the age of 14.
It was taking up too much time. - (So you do this instead?)
Well, I didn't start this 'till I was in grad school.
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