A question for mathematicians...

If you agree that the knight moves in a 2-1 pattern, where do you get 3 and 2 from. 

I think he means aS=1S+2S. So thats square root of 3.

The distance from a light-squred bishop to a dark square is infinte.

Well i dunno but what i think from the earlier posts ive just read is that it goes on a bit where the simple answer is (which they found out and then went on to argue something else (lol)) is that the diagonal is longer, so if the king moves at the same speed diagonally and horizontally, then the king will reach e3 first... because its shorter. So that means the diagonal to e6 is not the same length as the horizontal.

Wait here's an even shorter explanation: he says horizontal = veritcal = diagonal. WRONG. Diagonal is longer. Thats it. The end.

precisely.

You actually make an interesting point, the game of chess IS in two dimensions, obviously, but if we assume that as a graphical representation of actual action, say, in a battle, we would assume the pieces would have have the same speed in any direction, though pieces travel diagonals faster. The only way this could be described in a real-world situation may be due to some strange curavture in space-time. This doesn't imply that it would be in any dimension other than the 2nd, but that it is in a two dimensional plane contained in higher dimensions that the pieces are unable to comprehend that shape the movement of the pieces in ways that are unexplainable to a 2 dimensional being, we three dimensional creatures experience this as well, its all part of special and general relativity.

I disagree. Opposite colored bishops can never touch.  Think of it as a limit.

It's like dividing by zero. 1 / 0 isn't infinity, because 0 * infinity still isn't 1 (if you have 1 apple to divide over 0 people, even if you give all of them infinity apples, you'll still have the 1 apple left over).

In this case, it's not infinity, because even in infinite moves, the bishops won't reach each other.

Taking the limit would only make sense if the bishops would get closer to each other as you get closer to infinity. But they don't.

It's just undefined.

This is the thread that just keeps on giving...

The distance from a light-squared bishop to a dark square is always 1.

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edit:

I guess I do understand the implicit definition of distance being referenced was how many squares does a white bishop have to move to reach a black square. But you wouldn't define the distance the moon is from the earth as how many revolutions of the moon it took for it to reach the earth.

(And in the case of the moon that number would not be undefined.)

I guess we're all beginning to sound like Medieval philosophers.

Actually" a²=b²+c², correct, but you use the incorrect values for b and c

a²= 2²+1² = 5.

As far as the moon, the reason that its orbit would degrade so that it would eventually reach the earth I'm assuming might have something to do with thermodynamics and a decrease in order over time.  But even with chess, could we say it is inevitable that given enough time, a white bishop will reach a black square because inevitably someone will make an illegal move by accident or a cat will jump on the table bumping the white bishop, etc.  So we cannot expect the order imposed by an agreed upon set of rules to perpetuate for eternity.  So even distance measured in number of moves, the distance from a white bishop to a black square is neither undefined or infinite.

 The moon is actually moving away from the earth, a fact verified by the placement of mirrors on its surface by the Apollo missions, which are daily targeted with a distance measuring laser from the Earth. Current theory is that the moon was created when a Mars-sized object collided with the infant Earth in the early stages of the universe, throwing huge quantities of matter into orbit which coalesed into the moon. I think the point still stands though; the moon is moving away, but is, paradoxically, 'losing' energy due to tidal friction. The further out it's orbit, the slower it travels. I think. I might be wrong. Again

How in god's name did we get on to the orbit of the moon!?

But I thought all matter would eventually collapse into a point and then the Big Bang would happen all over again. (So the moon would eventually reach the earth).

That's a possibility, though scientists currently believe that the Universe will expand forever, though at the current point we are in we could be in any of the 3 states

I'm assuming you mean it will reach the earth if it continues to lose energy.

 The Earth/Moon system is independant of the Big Crunch (although current theory is that the expansion of the universe is in fact increasing, meaning the destiny of the universe is that all matter and energy would be too far apart to interact). Entropy is, on the whole, increasing, which suggests disorder is increasing, but this is frequently bypassed by open systems. Consider life. A living thing can be said to be an 'order creating machine'. This increase in order comes from the energy of the sun, which is slowly increasing in disorder over time, at a greater rate than order is being created in the form of life. Thus the total system is increasing in disorder.

I stand willing to be corrected.

*Mutters under breath - "chess is played in 1.5 dimensions"...*

 Apparently not. It's a paradox of celetial mechanics, but the moon is both losing energy and moving away from us, as it's orbital velocity is reducing. Consider the ice skater, who spins fast when he pulls his arms in, but slowly when he stretches his arms out.

If you were to go back in time a couple of billion years, the moon would have covered something like a quarter of the horizon. As I recall, it's moving away at a rate of a couple of cm a year.

I do see your point now.

When you move diagonally, you move both horizontally and vertically.  Say you could only perceive one dimension.  Whether it was the vertical or horizontal dimension you could perceive, if one square diagonally was moved, you would perceive a distance of 1 being moved (i.e. not 2^.5).  So I now fully see your point and think is is uncontroversial as far as it goes.  The rules of chess correctly perceive distance movement if that movement is directly horizontal or vertical.  If the movement is diagonal however, the rules of chess is only seeing one dimension (whether that dimension is the horizontal or vertical dimension is unknown and irrelevant.)  So from a given square, 4 of the eight squares to which you can move chess only perceives 1 dimension.  Thus chess perceives 1.5 dimensions.

However you did not consider the case of squares along the perimeter of the chess board and at the corners.  If a king is at A4 for example, only 40% of the squares it can move to are on a diagonal.  At A1 only 33.33% of the squares it can move to are on a diagonal.  So for all the squares on the board, the average dimension will be less than 1.5. You or someone needs to come up with the exact figure.