Saturday, August 6, 2011

CTY Challenge: Calculating pi like in the olden days!

This is the last week of CTY camp. Of course, a few activities deal with π because, who doesn't love pi? We did two activities that were attempts on calculating pi, with very little to use. Of course, we did the old make a circle and divide the circumference by the diameter. However, we also did a variation on the Buffon's Needle experiment.

The problem goes like this: take a ruler and draw a bunch of parallel vertical lines that are two inches apart and a foot or two long. Then, take a two inch needle and toss it onto the grid and determine if it crosses the gridlines or not. What do you think the probability is that the needle will cross the line?

If you do the work, you should get a number around 63.66%. However, try plugging in these values into this equation:

2t/c =

t = total number of tosses
c = total amount of times the needle crosses the line

What do you get? Pi, exactly. Our whole class tried it, and with over 1000 tosses, our class average is 3.35. Pretty good, right?

Our instructor never went over the proof in class, though it involves geometry, calculus, and more advanced statistics. I think it has to do with the center point of the needle, and where it is in relation to the gridlines. If you have a proof, please post it!

Additional Puzzle: A census taker walks up to a house, and records the house number. Then, he knocks on the door and a man answers the door. The census taker asks if anybody else lives with him. The man responds that he lives with his three children. The census taker then asks for their ages. The man responds, "Their ages add up to the number on the door and their product is 36." The census taker then says, "I need one more clue. Is the youngest child a twin?" The man says that the youngest child is not a twin. What are the kids' ages, and what is the number on the door?

Hint: Find all the possible combinations of three numbers multiplied together to get 36 (including ones where the smaller numbers are equal).

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