My name's Philip Ripper. I'm not a student. I'm a curious person with a fairly poor math education. This is not a homework question, but that aside, I will try to show the effort I've put forth to solve it.

The problem comes from a board game called Blood Bowl, a fantasy wargame simulation of american football (it's great fun).

There is an action in the game called a Hail Mary Pass, HMP for short. When you do this, you throw the ball at a location on a grid. The ball is never accurate, and Bounces three times before coming to rest (at which point someone can try to catch it).

Bouncing: label the 8 squares surrounding the starting square from 1-8, and roll an 8 sided die to determine where to move the ball. The ball Bounces three times. (So think of it as a 3x3 grid, with the starting square in the center, the ball bounces to a random square not including where it started for that bounce - 8 possibilities per bounce).

I'm trying to figure out the odds that the ball will land on each possible grid square after a HMP (three bounces). I figured this creates a 7x7 grid of possible locations.

I know 8x8x8 = 512, so there are 512 possible dice combinations. And I know that the four extreme corners of the grid (3,3; 3,-3; -3,3; -3,-3) are each a 1 in 512 chance.

I know to figure the odds you are supposed to determine the number of events (512) that create each outcome. I also know the problem of bouncing a ball three time sounds simple, but that once anyone I know tries to solve it they give up.

I think it might be considered a markov chain? From trying to google for a formula that I could understand to find the solution without just counting out each of 512 events / 49 states manually.

I've worked in a spreadsheet (I'm not good with those) for about two days (as in 48 hours of work) trying to solve this problem. I've come close several times, but always end up with a probability total of more than 1, so I know I'm always wrong.

I charted out the options of the d8 roll as:

1: +1x +1y

2: +1x +0y

3: +1x -1y

4: +0x +1y

5: +0x -1y

6: -1x +1y

7: -1x +0y

8: -1x -1y

And considered the starting point as 0,0. I think this would allow me to brute force the problem, but I really don't want to count out 512 variations (though I would have been done by now if I had, I guess).

Is there a way to solve this that a high school drop out could understand? I'm afraid I don't understand the formula for markov chains, if that is what this problem is.

If there is no conceivable way for a guy with a spreadsheet to solve this without understanding complex mathematics, could anyone solve it for me?

I'd prefer to understand how to solve it. But I would still be happy with an answer instead, because I fear I may just not be able to understand it. Though I am eager to find out if I can. I've tried very hard to figure this out with the tools available to me. Please assist!

Sincerely,

Philip