During my cross-country bike trip with two friends, I ran into an interesting spot at an Italian restaurant in Murphysboro, IL. We ordered a large portion of toasted ravioli as an appetizer to be split among us. The ravioli were of different fillings and different size.
What we ended up doing as tired cyclists was haphazardly eat away at whatever was on the plate, but I can't help but wonder: what if this were something valuable, like a large inheritance, or a huge plot of land?
Surely minor differences in toasted ravioli crumbs can be forgiven, but what about discrepancies that could be potentially thousands of dollars?
You split, I choose.
Most people are familiar with how to do the split between two people. It's pretty straightforward. The so-called divide and choose method of splitting something dates back to the start of time.
Literally. In Genesis Chapter 13, Abram splits the land of Canaan into two, and asks Lot to choose a piece.
So Abram said to Lot, "Let’s not have any quarreling between you and me, or between your herders and mine, for we are close relatives. Is not the whole land before you? Let’s part company. If you go to the left, I’ll go to the right; if you go to the right, I’ll go to the left."
The logic behind the method is straightforward:
- As a divider, all parts are considered equal. Regardless of what the chooser decides, the divider receives a fair portion.
- As a chooser, the piece chosen must be as least as valuable as the other.
So, if two is straightforward, so must three. Right? Not quite.
What about for three?
Here is the fairest way to split anything among three people:
|The first person divides the cake into three pieces that he or she considers equal.|
|Then, the second person considers the biggest two pieces. He or she cuts the biggest piece so that the two pieces are equal.|
|The third person chooses a piece.|
|The second person chooses a piece, but if the piece that he cut is available, he or she must choose it. (Shown here)|
|The first person chooses the remaining piece.|
|Between P2 and P3, the person that did not choose the twice-cut piece divides the remaining piece into thirds. (P3 here)|
|The person that divides chooses last. P1 chooses next. The remaining person chooses last.|
|And finally... we're done.|
The procedure was not invented until 1960, when American mathematician John Selfridge conceptualized the entire process.
Not until 1993, over three decades later, did John Conway, British mathematician and creator of Conway's game of life, independently come up with the procedure and take half the credit for it.
Today the Selfridge-Conway discrete procedure is the gold standard for dividing anything into thirds with at most five cuts.
In the interest of time, I leave it to the reader to figure out why it's fair.
Easy for two, hard for three.
As many of you may know, this is the quadratic formula.
Straightforward enough to be featured as a middle school song, the quadratic formula is capable of solving polynomials of degree two. But what about for three?
There are countless examples in mathematics, physics, biology, and chemistry. The two-body problem, determining the motion of two objects that interact with each other, is trivial. The three-body problem? A field of active research.
Bisecting an angle with compass and straightedge? Easy. Trisecting it? Impossible.
In topology, there are many examples. Proving "if every loop on a 2-d surface can be tightened to a point, then we have a sphere" is fairly straightforward.
For 3-d surfaces? Well, that's enough to win a Fields medal, solve one of the Millenium Prize Problems, and be awarded "Breakthrough of the Year" by Science.
A Whole Journey Lies Ahead
There's more to this article than splitting some toasted ravioli. The broader field, envy-free division, is a field of active research and has applications ranging from video games to corporate acquisitions.
What about for four or more people? Well, it turns out, there are whole university courses and books written about the subject. One such book, written by Drs. Steven J. Brams and Alan D. Taylor, forms the basis of the Brams-Taylor procedure, the first division procedure for an arbitrary amount of players.
So yeah, the next time you need to split something between three people, just say "you split, I split, he chooses, I choose, you choose, I split, you choose, he chooses, and then I choose."
Practice makes perfect.