Pi Day

Pi day was . . . last Thursday. I had raspberry pie to celebrate. I also spent two and a half hours at RPCC for dinner, but that’s another story.

If you don’t know me or haven’t been reading my blog, you might not know that I like math. Unfortunately, I don’t have any math planned for today. Fortunately (or also unfortunately), I can just make it up as I go.

So, here goes. The closest fractional approximation for pi with both numerator and denominator under one hundred is 22/7. This comes out to be 3.142857, repeating all six decimal places. Sometime in middle school, I learned that every fraction can be written as a terminating or repeating decimal. Conversely, any repeating decimal can be written as a fraction (being able to write terminating decimals as fractions is fairly obvious).

Since then, I have probably gotten way too much amusement dividing fractions out mentally until they repeat. My personal favorite is 1/7, which happens to be the fractional leftovers of 22/7. Which brings us back to pi day.

Well, pi sounds like pie, which is like cake, which leads me to another fun mathematical problem. Suppose you have a square cake with icing on its top and all four sides. Suppose you also have ten friends who all want slices of cake with not only an equal amount of cake but also an equal amount of icing. Suppose any leftover cake will be smashed into your hair by angry friends who are mad that they didn't get more cake (this will also happen in the case of unequal slices). One cake, eleven people – how do you slice this cake to avoid washing dessert out of your hair?

It turns out that by dividing the perimeter of the cake by the number of slices you need, marking out these divisions on the cake perimeter, and cutting from each mark to the center, every slice will have the same amount of cake and icing. Seen from the top, each slice can be thought of as a triangle (or two triangles, for pieces spanning the corners). All the slices have the same height, as well as the same base (sum of the two bases for corners), since the bases are equal portions of the perimeter.

Therefore, since the area of a triangle is half of the base times the height, every person gets the same amount of cake and icing. Other solutions include buying a circular cake (you’d naturally divide a circular cake in exactly the same method as described above, but for some reason people like cutting square cakes into square/rectangular pieces), tossing the cake into a blender and distributing portions by weight, or getting rid of three friends (cutting eight equal pieces is a lot more intuitive than cutting eleven equal pieces).

I read about the solution (the first one) to the cake problem in the book Why Do Buses Come in Threes? (if you couldn't tell, the last three solutions are mine). This is one of my favorite books, and it discusses how math is involved in everyday life. Yes, I read books about math in my free time.  Credit to the authors for compiling the mathematical solution, but also feel free to use any of mine.