A chess board riddle

[Yoni]

Given is a standard 8x8 chess board. A regular chess rook is placed on one of the corners.
(Reminder: A rook travels in straight lines only.)

The rook needs to travel to the corner that is diagonally opposite from where it started.
On its way, it has to visit all the squares on the board, and each square exactly once.

Can this be done?
If so, give an example of such a trail.
If not, prove it.

[Soul Taker]

I can touch all squares except one.  There's two mutually-exclusive squares at the end in my closest solution 

[Meh]

Are all the other peices on the board? If so how do we know where each peaice has gone? If not then go up and down in a line.

[l)ragon]

If you go by the fact that you dont actualy visit the square untill you place the piece on it, this is your answer in black text.

S = Start, F = Finnish
when you get to 1 you move to 2 then to 3 and follow to the end.
X > X   X > X   1   3   2   F
^   v   ^   v   ^   v       ^
X   X > X   X > X   X > X > X
^                            
X < X < X < X < X < X < X < X
                            ^
X > X > X > X > X > X > X > X
^
X < X < X < X < X < X < X < X
                            ^
X > X > X > X > X > X > X > X
^
X < X < X < X < X < X < X < X
                            ^
S > X > X > X > X > X > X > X

Pointing out you can do this with the knight which moves in an L shape aswell and touch every square.

Edit: blackOtherwise its not posible just try it on a 2x2 or 4x4 board, you will allways have 1 square that you cant get to./black

[hismajesty]

Does it have to start where it would in a normal game?

[Yoni]

Are all the other peices on the board? If so how do we know where each peaice has gone? If not then go up and down in a line.

No. There are no other pieces on the board. This is not a chess game, it's a math riddle.

dRAgoN's solution is creative, but incorrect.
You are visiting the square you marked "3" twice. Yes, it counts as a visit if you just "go through it" without stopping.

Does it have to start where it would in a normal game?

Well, this isn't a "normal game" - it's not a chess game, it's a math riddle that uses a chess board. You have a rook on one corner, and no other pieces.

[hismajesty]

So the answer is yes then?

[Yoni]

So the answer is yes then?

I guess so... But the only things here that resemble chess are the board, and the way the rook moves.

[hismajesty]

And where the rook starts off.

[Yoni]

Since nobody solved it so far, I will give a hint in color=black - don't read it if you don't want a clue.


As many of you have already guessed, it is not possible for the rook to trace a path as asked.
A few reasons for this:

1. You've worked on the problem for 10, 20, even 30 minutes, and still haven't found a path.
2. The question is on the math forum, so it must involve some math. If there was a path, the problem wouldn't be so interesting, since it's not mathematical enough for this forum, I think.
3. Because I just said so. Trust me, this is true.
4. Because there is a mathematical proof that proves there is no path.

As a continuation to reason #4, you are now challenged to prove it.
Guidelines for a proof:
A. The proof is not complicated. (As in, if I tell you the proof, you will understand it.)
B. The proof does not use high level mathematics. (If you try to prove it, and you start bringing in calculus, you're going too far.) There are no equations - simple logic deductions are sufficient. However, the proof is mathematical, not "I tried a lot and it never works".
C. The proof is elegant - short and to the point. (Don't make a table of all the possible moves for the rook please.)

Good luck!

[hismajesty]

I think it's not possible because I tried a lot and it didn't work and also because the number of squares is even!!!

[l)ragon]

odd: x=y²
even: x=y²-1

[Yoni]

odd: x=y²
even: x=y²-1

What are x and y and what are you talking about?

[l)ragon]

x is the number of squares your able to visit
y is the width of the board

[Yoni]

Care to prove this?
Edit: