# Wild About Math!Making Math fun and accessible

13Oct/080

## MMM #17: The bishop and the king

Doug Hull was randomly chosen as the winner for MMM #16 over at Blinkdagger. Congratulations, Doug!

It's time for a new contest problem. As you may have noticed, I like counting and probability problems. I found an elegant one in an old book. I won't reveal what the book is until the end of the contest. Here's the problem:

If you randomly place a bishop and a king on a normal 8x8 chessboard, what is the probability that the king will be in check?

For those of you who aren't chess players, here's an equivalent problem description: If you randomly select two squares on an 8x8 chessboard, or checkerboard, or simply in an 8x8 grid, what is the probability that the two squares will share a diagonal?

For special recognition, but no increased chance of being selected as winner, generalize your solution for an MxM chessboard, where M is even. For extra special recognition, generalize your solution for an NxN chessboard, where N is odd. And, for super duper extra special recognition, come up with a general solution for an AxB chessboard, where A and B are arbitrary integers, both greater than 0. I've not explored the AxB generalization and it's not mentioned in the book that I got the problem from; it might be tough but I believe the problem and the three generalizations will make for great mathematical exploration.

I have a Rubik’s Revolution, courtesy of Techno Source (or \$10 Amazon.com gift certificate) to give to the winner. I’ll give more than one prize if I get lots of correct submissions.

Here are the rules for the contest: