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.

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

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.

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.[color=Black][pre]
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
[/pre][/color]
Pointing out you can do this with the knight which moves in an L shape aswell and touch every square.

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

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

[quote author=(V)eh link=topic=8983.msg83019#msg83019 date=1096803589]
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.
[/quote]
No. There are no other pieces on the board. This is not a chess game, it's a math riddle.
[hr]
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.
[hr]
[quote author=hismajesty[yL] link=topic=8983.msg83039#msg83039 date=1096820313]
Does it have to start where it would in a normal game?
[/quote]
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.

So the answer is yes then?

[quote author=hismajesty[yL] link=topic=8983.msg83052#msg83052 date=1096823029]
So the answer is yes then?
[/quote]
I guess so... But the only things here that resemble chess are the board, and the way the rook moves.

And where the rook starts off.

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.

[color=black]
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.)
[/color]
Good luck!

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!!!

odd: x=y[sup]2[/sup]
even: x=y[sup]2[/sup]-1

[quote author=dRAgoN link=topic=8983.msg83134#msg83134 date=1096852873]
odd: x=y[sup]2[/sup]
even: x=y[sup]2[/sup]-1
[/quote]
What are x and y and what are you talking about?

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

Care to prove this?
Edit: :P

[quote author=Yoni link=topic=8983.msg83153#msg83153 date=1096861650]
Care to prove this?
Edit: :P
[/quote]
Do you realy think anyone would paste '33,439,123,484,294' diagrams to prove this lol ;p

No. See guideline "C" in my blacked out hint.

This sounds like that proof Euler did on the bridges in that town.  Graph theory and what not.  We talked about it in my algorithms class.  T'was so long ago, so I don't remember much.

Over 140 thread views and still no solution. Oh well. I will post the proof here for anyone who's still interested.

[hr][color=black]First, paint the squares in the chess board black and white.
Do it like a regular chess board.

Observe the following property:
1. The starting corner and the ending corner of the rook have the same color.
(If the rook started on a white corner, it must travel to the other white corner, and if it started on a black corner, it must travel to the other black corner.)

Also note:
2. The rook must make 63 steps in its journey.
(It has to go through every square once. It's already in the first, so it must make 63 moves.)

Now, let's look at the properties of a move:
3. When a rook makes 1 move, it always moves to a square of a different color than it is now.
Simply because any adjacent square to a white square is black, and vice versa.

Now, to further generalize property #3, observe the following sequence of moves:
Let's say the rook starts at a white square, and starts a journey.
Then it travels through squares of the colors: Black, White, Black, White, Black, etc...
It's no problem to see (and prove by induction, too) the following:
4. When a rook makes an odd number of moves, it always moves to a square of a different color than it is now.

Now, knowing the above properties, we can ask: Can the rook make its journey?
5. The rook cannot make its journey. If it were able to, it would make 63 steps (according to #2), therefore end up in a square of a different color than it started with (according to #4), which cannot possibly be the ending square (according to #1). Q.E.D.[/color][hr]
Now that you know the solution, attempt my other riddle! :)
https://davnit.net/bnet/vL/phpbbs/index.php?topic=8999.0