OR: What happens when you don’t define the shape of your pie

Last month heralded Pi Day, in celebration of the mathematical constant and irrational number Pi.

To mark the occasion at here at Digital Science, we held an informal math challenge. Don’t be scared to keep reading if, like me, mathematics isn’t your first language. The only equations here involve Justin Bieber, pie filling, and tenuous but amusing proofs.

640px Pi Pumpkin Pie, January 2008

The challenge was as follows:

“You have exactly 400 x Pi cubic centimetres of the finest pie mix [about 1,250 cm cubed], but you don’t have much pie crust mix and you want to use the smallest amount possible. What are the best dimensions to use for your pie, to use the smallest amount of crust? Hint: check out the SureChem logo. Double hint: look at both the shape and purpose of the logo.”

…and the prize at stake was various tasty pies from a local bakery.

For the more math-savvy, the problem isn’t too hard to work out – it’s an optimization problem, and it can be solved using some calculus.

Unfortunately, maths problems require enough information so they can be solved correctly, and despite a few hints pointing to the shape of the SureChem logo (a Digital Science product), it quickly became apparent that there wasn’t enough information about the shape of the pie. The resulting interpretations of the problem from our colleagues were far more interesting than the actual answer.

Answers included:

A fairly logical conclusion: “At the moment the answer is that the pie is spherical with radius 6.69433cm (or 2.13Pi cm, if you like).”

A serendipitous and delicious invention: “Assuming a [company] canteen-style pie, where you basically get some pie mix, add mystery meat, and then top it off with a crust. So, the problem becomes that of minimising the area of the crust on top. Clearly then, the best pie shape is an infinitely long cylinder with an infinitesimal speck of crust on the ‘top’ – although I guess it’s a philosophical question as to whether an infinitely long cylinder has a top. This will not only be delicious, if we spin the cylinder it will also act as a convenient time machine.”

Some helpful housing advice: “Alternatively, create a spherical shell of pie filling, with the hollow centre as small as possible. Define yourself to be inside the pie, making the hollow centre the outside and coat that in pie crust. Depending on how dextrous you are, the amount of pie crust required can be infinitesimally small. This method shows how you can reduce the costs of living by careful use of pure mathematics.”

A practical refinement: “Your infinite pie crust solution doesn’t work, because to be pie crust, there necessarily has to be things that make it pie crust, i.e., flour, water, butter, or a.) it will have to be made of some exotic matter, and b.) it will be too soggy. In order to minimise pie crust area while still getting a delicious golden brown finish, we will stack these ingredients in the correct ratios in a pile one atom across. Therefore the pie lid will be one chlorine diameter across as I think chlorine in the pinch of salt is the biggest atom in pie. My periodic table app keeps force closing.”

Also an idea for a Stephen Hawking-inspired episode of a home improvement show: “Alternatively, you could set up a kitchen inside of a black hole, in which your pie mix will be compressed to an absolutely minuscule volume. You then have to spread the pie crust mix as thinly as possible to cover everything. If it still isn’t enough, you just move deeper into the black hole. There’s one caveat though – you have to consume your entire pie inside the black hole as well – if you wanted to bring it back out of it (which is no mean feat I tell you) – the entire pie will explode…”

A few observations about the precise definition of a pie cropped up too. One colleague was keen to point to the Wikipedia page for pie, which indicates that pies are “defined by their crust”, thus rendering the entire competition null and void because critical information was omitted from the description. It was also suggested that the shape of the pie in the correct answer was actually more or a “deep-pan tart”. Finally, one helpful insight was that “if they can get away with making “beef” lasagne with 100% horse, I’m sure we can make a pie (of any shape) without using any actual crust.”

There were also cartoon and picture submissions including The Wife of Pi, The Life of Pi, and something of a wierd coincidence.

The pie-winning entry, for submitting the correct answer before anyone else and for the funniest submission came from a member of the Digital Science finance team. The correct answer is a pie with both radius and height of 7.368cm, which was skillfully written up with the help of Justin Bieber and writelatex in a groundbreaking mathematical proof.

If you’re wondering where the correct answer comes from, check out this video
by Jim Fowler of Ohio State University. It’s for the “soup can” problem, which the pie problem is closely based on.

Thanks to everyone for submitting their answers, and to Jim Fowler of The Ohio State University and Coursera, both for running an amazing Calculus course and for providing the inspiration for the Pi-day challenge.

Lesson learned: Always define the shape of your pie.

Attribution: Pi-pie photograph by Paul Smith.