SRU math major flips over pancake problem


Pancake on a plate

Dec. 5, 2017

SLIPPERY ROCK, Pa. - Imagine yourself standing in front of a hot griddle, the scent of pancakes wafting into the air. Now picture those pancakes stamped with a smiley face on one side and a frowny face on the other.

As you turn the pancakes over with an oversized spatula, you wonder how many flips in the air it would take until all five pancakes landed smile-side up.

While you're still standing at the griddle, Gennifer Farrell, a Slippery Rock University dual mathematics and philosophy major, has already discovered the answer. All four answers, actually.

Farrell, a senior from Carlisle, answered the call when Google poised the pancake dilemma at part of its Code Jam 2017 during its 14th annual competition in Dublin, Ireland. While she did not participate directly in the Jam, she decided to solve the competition's problem statement in her free time.



The problem was accompanied by the following circumstances:
-A chef is cooking pancakes arranged on the griddle in a single line.
-These pancakes have two sides, happy side up and sad side up.
-The chef does not have a spatula that is capable of flipping one pancake at a time.
-He only has a spatula that can flip a fixed number of two or more pancakes at a time.
-His spatula is also not bigger than the total number of pancakes that need to be flipped.

The challenge for Farrell was to calculate the following:
-Determine - given any number and initial orientation of individual pancakes, and given any valid spatula size - whether it is possible that the pancakes can be flipped so that they are all oriented happy side up.
-Given that it is possible to flip all the pancakes so that they are oriented happy side up, the minimum number of flips necessary to accomplish such a feat must be found.

"To solve how many flips it would take to get all five pancakes facing happy side up, I didn't really use any math you would normally come across in the classroom," Farrell said. "I mostly relied upon concepts I learned in number theory and proof-based math classes to achieve my answers."

Using the symbols "p" for pancake and "s" for spatula, Farrell entered each possible combination into various software programs to determine whether the pancakes could eventually face happy-side up at the same time.

The next step was to determine the minimum number of flips to reach the proposed solution. For this, Farrell utilized a unique mathematic concept called modulo-2 arithmetic. She compared this sort of math to the cancelling out techniques one would use to solve a Rubik's Cube.

"There are going to be a lot of combinations possible for five pancakes. In fact, I ended up with 32 different combinations that are possible," Farrell said. The solutions Farrell came to took approximately one month for her to determine.

Once she solved the problem, she submitted her work and had it accepted for presentation at MathFest 2017 in Chicago, an annual conference facilitated by the Mathematical Association of America. The MAA is the world's largest community of mathematicians, students and math enthusiasts. The mission of the MAA is to advance the understanding of mathematics and its impact on the world.

"It seemed to garner a lot of positive feedback. I was there totally out of my depth, but a lot people really seemed to enjoy the problem."

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