witzar

A problem is similar to (see my solution). You are adding new point on the circle and drawing new chords. For each chord you count intersections with old chords and add 1, this gives you the number of new regions. This is how you arrive at the formula. Last step is to simplify it. But to be honest I just guessed the solution I took the formula from above mentioned problem and made an adjustment for that extra 1s. Few tries and it worked. Edit: Guessing helped me only in the last step (simplification). I needed non-simplified formula to verify guessed one.

Anyone cares to show why the other two solutions I've found (in post #5) are wrong?

Commutativity of multiplication makes it irrelevant, I believe.

It couldn't find 10x10 square.

More squared bishops in attached file. bishop.txt

They do of course. Bushindo made a silly error evaluating your formula.

That's the obvious and natural approach. And that's exactly how I got the formula. When you say "close but", you mean that the formula is wrong?

Is the road a straight line?

If you don't like Mr Bayes, you can draw a picture. Now all you have to do is divide green by (green+red).

http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0

OK. But why all squares of the chessboard are of the same color on the picture?