## 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:

- Email your answers with solutions to mondaymathmadness at gmail dot com.
- Only one entry per person.
- Each person may only win one prize per 12 month period. But, do submit your solutions even if you are not eligible.
- Your answer must be explained. You must show your work! Wild About Math! and Blinkdagger will be the final judges on whether an answer was properly explained or not.
- The deadline to submit answers is Tuesday, October 21, 12:01AM, Pacific Time. (That’s Tuesday morning, not Tuesday night.) Do a Google search for “time California” to know what the current Pacific Time is.)
- The winner will be chosen randomly from all timely well-explained and correct submissions, using a random number generator.
- The winner will be announced Friday, October 24, 2008.
- The winner (or winners) will receive a Rubik’s Revolution or a $10 gift certificate to Amazon.com. For those of you who don’t want a prize I’ll donate $10 to your favorite charity.
- Comments for this post should only be used to clarify the problem. Please do not discuss ANY potential solutions.
- I may post names and website/blog links for people submitting timely correct well-explained solutions. I’m more likely to post your name if your solution is unique.

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