Arcs in a Square (or Snakes on Plane)
In a recent MAA publication, Shai Simonson, attempts to bring the joys and excitement of the world of mathematics to the non-technical reader. In Rediscovering Mathematics: You Do the Math, Simonson covers a wide array of topics ranging from number theory to the application of probability in sports, casinos and gambling. I have added the book to my reading list and you might want to take a look at the article that begins his text, “How to Read Mathematics.”
One of the problems from the book was posted over at Math Mama Writes… and when a puzzle like this piques my interest, I’m at its mercy until I figure it out. Thanks to a recent illustration I made in calculus last week and a cartoon that reinforced my perspective, I have a new motto for next year’s courses:
Math problems aren’t solved, they are conquered!
Well, this problem below was one that I had to defeat. I went to battle with it and after losing a few skirmishes (i.e., trying approaches that failed) I finally beat it into submission. From now on, when I see that feisty integral that won’t behave or a simple number puzzle whose pattern defies identification, I’ll strap on my armor (or sweater vest), grab my sword (or calculator) and wage full-out war on that problem. No problem is safe!
Arcs in a Square (or Snakes on a Plane)
Given the square ABCD, with side length 4 and circular arcs centered at each vertex, find the area of the region at the center - without using calculus.
And by the way, the Snakes on a Plane reference is my own and I’ve not ever heard this problem referred to in this way but it sounded good to me (arcs 
snakes, square plane). However, I am a strange duck.
I’ll post a couple of different approaches that conquered this problem in later posts.