Math Delights

G'Day All:

This is the final St. Mark's Institute of Mathematics e-mail for the year. Students at St. Mark's School have already left for the summer and teachers and administrators are starting to relax and take it easy too. BUT ... there is time for me to sneak in one final mathematical teaser as a send-off for the summer. This next puzzler is a variation of the famous "Buffon Needle Problem." I've added a devilish bent! Here goes ...

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WOODEN FLOORS AND BENT NEEDLES

I have a wooden floor composed of uniform wooden slats each one inch wide. Thus the gaps between the slats make a series of parallel lines spaced one inch apart from each other. I have a wire needle one-inch long and I shall throw it on to the floor. But before I do so, I bend it.

a) [WARM-UP] Suppose I bend the one-inch wire needle into a circle and then throw it on to the ground. What is the probability that the circlular needle lands touching or crossing a gap between two wooden slats?

b) Suppose instead I bend the needle into a perfect square, each side one-quarter of an inch long, and then throw it onto the floor. What is the probability that the square needle lands touching or crossing a gap between two wooden slats?

c) Suppose instead I bend the needle into an equilateral triangle before throwing it on to the ground. What is the probability that it lands touching or crossing a gap between two wooden slats?

COMMENT: Parts b) and c) are hard, unless you can come up with a philosophical reason why their answers must be ...
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I will accept solutions to this puzzler all the way up until September 1, 2010. So feel free to take your time to mull and reflect. Parts b) and c) are best thought about while walking, or jogging, or climbing a tree. So my advice is to go outside and enjoy summer fun - while thinking math!

I will offer a pan of Dr. T's absolutely should-be world famous double chocolate coconut fudge brownies to the person who has the most success with this puzzler.

Enjoy! - and Happy Summer!