Interesting Chessboard Math Problem

Two pawns on opposite sides of a 64 square, standard chessboard. Every second, the pawns travel from the square they are on, to another adjacent square, which is chosen at random.

So, here are the questions:-

How many second would it take until the pawns both sit on the same square on the chessboard?

How long until the pawns cross paths with one another, i.e., if pawn 1 is going from A-z and pawn 2 is going from Z-A?

Is the first a minimum of five seconds and the second a minimum of four?

the answer can be quite odd because of course there is a probability they will never meet. may i ask where you found this question?

What's the initial set up exactly? can you give there starting squares in chess notation?

What's the initial set up exactly? can you give there starting squares in chess notation?

Just a thought here, haven't really put much into it and haven't really done much stochastics (or combinatorics?) before. Would it be simpler (or even the same problem?) to say fix one of the pawns and assume the other one moves 2 squares each second and ask the average time for it to end up in the opposite corner?

As it's hard to describe the movements on a board in words I will assume you are looking at a board like you were playing chess and the pawn is in the left corner on your side and you're wanting to get it to the right corner on the other side.

I'm going to call an "up step" one step away from you and down one towards you. Left and right steps are obvious.

The problem is then what would be the expected time for the number of up steps = number of down steps + 7 and number of right steps = number of left + 7.

Then I don't know

Well, I've found what seems to be a bug in Mathematica's treatment of DiscreteMarkovProcess while trying to investigate the problem. That's as far as I've got.

Just a thought, if you were to try to solve this problem instead for a general nxm board, couldn't you get a reccursive relation by conditioning on the first move?

Though a 2 variable recurrsive relation may not be much use for calculation purposes if it cannot be massively simplified.

Yes, you could, but it would be really quite disgusting. I briefly contemplated setting it up, but decided not to touch it with a barge-pole.