A question for mathematicians...

How do you know the knight is taking the shortest path? Some beginners go "one two over" and so the knight might travel more.

:-)

The discussion here is interesting.  :)

As for the distance the knight moves, if you use the distance formula d = SQRT ((x1 - x2)^2 + (y1 - y2)^2) then you get d = SQRT (5).  The squares on the chessboard have coordinates (x, y) where x represents the file and y represents the rank of the square.

And there you have it. Einstein is never wrong as long is religion is not involved. 

Thanks 

This discussion is completely meaningless. Anyone who believes "chess is 1.5 dimensions", please give a complete, rigorous definition of "dimension", and then we'll talk.

The distance formula, The Square root of the quantity ((x)^2+(y)^2). Therefore, you can go any # of "spaces" or "squares" you should like, but the distance remains the same. Also, the king moves further on diagonals lol, It moves the distance of 1^2+1^2=2 while on a line it goes 1^2+0^2=1. Therefore, it is played on 2 dimensions, by definition BECAUSE the x,y cooridnate plai still applies. The logic is flawed also because there is no such thing as a half dimension. The plains are x,y,z. You can't have a half of an x-plain. That's just stupid. No offense intended of course. If you don't apply any math your reasoning is sound.

1 is easy.

maybe we should discuss displacement instead of distance :)

This is the Chebyshev distance that you are talking about.  :)

http://en.wikipedia.org/wiki/Chebyshev_distance

your logic is all messed up.  the triangle can't include the square that the king is actually on. and also the distances ARE the same three units=three units 3=3 in any dimension that I am aware of.....

There's something satisfying about starting a thread that captures people's imagination and gets very long. I wonder what other mathematical curiosities I can post up...

Posting one mathematial curiosity like this per week should make the day for everyone...atleast for people like me who despite knowing the abstraction level of this thread have still followed it religiously for the past 2-3 days... Is there a deadlier combination than math and chess...

Cheers,

Arun

 I'm working on it. Watch this space!

As the resident MIT math guy, let me see if I can answer the original poster's question.

Math is nothing more than a convenient simplification of reality. Everyone knows that chess is REALLY four dimensional (three dimensions of space and one dimension of time) because those are the dimensions in which we live our lives, but it's convenient to think of chess as two-dimensional. The height of the chess pieces, for example, has no effect on the outcome of the game, so we can safely ignore the third dimension of space. Likewise, whether you take a minute or an hour to decide your next move, it doesn't really change how good that move is. Time controls are just so that we can get home in time for dinner, not something inherent to the game itself. So, we can ignore the fourth dimension of time as well.

What we are left with is a two-dimensional game. There is nothing magical about two dimensions, it's just convenient. Especially with the advent of algebraic notation. A game can easily be expressed in two dimensions, and can easily be played in two dimensions on a computer screen, or written about in two dimensions on a sheet of paper.

However, this does NOT mean that chess "is two-dimensional." Whatever that means. Again, the dimensions are just things we made up as a simple model to make things easier to understand.

Chess, believe it or not, can easily be expressed as a one-dimensional game. Instead of having a square board, I could have a simple number line from 1 to 64. I would have to translate the movement of the pieces into a slightly complicated set of rules, but it would work. A pawn on square nine, for example, could move to square 17 or 25, and could capture on square 18. The only reason we don't play chess in one dimension is because it's inconvenient and confusing.

A chessboard could also be described using polar coordinates. It would be weird, since the board is square and not around, but of course it could be done.

The reason the original question is slightly confusing (and very interesting) is because he's trying to force the rules of chess into a Euclidean geometry. But, the rules of chess have nothing to do with geometry, or triangles. In fact, the rules of chess have nothing to do with MOVEMENT or even DISTANCE. The only reason pieces move at all is because that's how you physically get a piece from one square to the next. Pieces don't have to move on a computer screen, for example, they can just blink in and out. The board could be made rectangular, as another example, which not only would make diagonals have different lengths than files and ranks, but now suddenly e4-e5 is a different length than e4-f4.

It's important to remember that chess is a game based on rules, not dimensions. The fact that we use two dimensions is simply because it's an easy way to visualize the rules.

Check out http://www.philipbrocoum.com/?page_id=91 for some more of my math explanations.

pbrocoum - nicely out. If I understand you correctly, are you saying that applying dimensions at all to chess is merely for our convenience, and that the only thing that is inherent to chess are its rules, which are independant of dimensions?

I would contest that we can ignore time, though. We can ignore actual time, which, as you rightly put, has no real bearing on the game. My measurment of time was the tempi, where one piece moves per tempi (strictly speaking, one piece moves per half-tempi). With this in mind, would it not be accurate to say that a King moves at a speed of '1 square per tempi'?

A single board configuration in chess is different from a game of chess.  No game of chess can be accurately characterized without reference to the dimension of time. The time a piece was at a given square is what is relevant.  That time would be expressed as a move number relative to the beginning of the game.

The King moves two squares at a time when moving diagonally. It would take a rook two moves to travel the same distance.

Chess in its current state is in two dimensions, it COULD be reduced to one dimension, but then a diagonal move would take the king a long way across the board. So, in 1-D chess, the king can hop all around the board by changing files. However, the rules were designed for a 2-D board. So arguably, the king still moves a distance of two squares in a diagonal move no matter what dimension it's in. On a 2-D board, it looks like one square, on a 1-D board, it looks like nine.

As for time, it currently has no affect on chess, but just as you could move chess down a dimension, you COULD move it up one, or two. In 4-D chess, time would have an affect on the pieces.

Well if "move" is defined as "to change location"  I don't know how chess has nothing to do with movement.

I was sort of hoping for an MIT guy who would dazzle us with bizarre perspectives, not just demystify everything.

 Physics, history and chess...