Egyptian Fractions

Rhind Papyrus: © Trustees of the British Museum.

How would you divide 8 cupcakes equally between 5 people?

The answer we learned in elementary school is to give each person \(\frac85\) or \(1\frac35\) of a cupcake. But, practically, how do we cut a portion that is precisely \(\frac35\) of a cupcake?

One solution is to use an iterative process. To begin, you have more cupcakes than people, so you give each person 1 cupcake and then you’re left with 3 cupcakes to divide among 5 people. You can easily cut each in half, so you have 6 halves to distribute. Of course, this means that each person gets a half and then there is 1 piece leftover which can be further divided into 5 equal pieces (or eaten by the cake cutter when nobody is looking). These last pieces are a fifth of a half - or \(\frac1{10}\) of a cupcake.

So we have taken the 8 cupcakes and given each person 1 cupcake, then \(\frac12\) a cupcake, and finally \(\frac1{10}\) of a cupcake. Since everyone got the same amount of cake with no leftovers, we see that

\[\frac85=1+\frac12+\frac1{10}.\]

An infographic showing this process can be seen at this link.

This kind of decomposition is called an Egyptian Fraction Decomposition (EFD) and it was first seen on the Rhind Papyrus. This papyrus dates to approximately 1550 BC and was found near Luxor, Egypt in the 1800s AD. It contains tables of Egyptian Fraction Decompositions as well as methods for decomposing fractions and some problems about dividing a specified number of loaves among 10 men.


The method we used to divide the cupcakes resulted in smaller and smaller fractions with a 1 in the numerator and each subsequent denominator dividing the prior one. But why couldn't we have just said that \[\frac85=\frac15+\frac15+\frac15+\frac15+\frac15+\frac15+\frac15+\frac15?\]

Well, not only is this not the easiest way to divide up the cupcakes since it requires each to be split into 5 equal pieces, but it violates the rules of EFD. Mathematics is all about taking a set of rules and seeing what questions you can answer within that framework.

Okay, then lets list the rules for an Egyptian Fraction Decomposition.

• The terms of the decomposition must have a 1 in the numerator.
• Each term must be different from the other terms.
• Infinite sums (what mathematicians call a series) are not allowed, so there must be a finite number of fractions in the decomposition. 

Then we can see that \[1=\frac12+\frac12\] and  \[1=\frac12+\frac14+\frac18=\frac1{16}+\frac1{32}+\frac1{64}+\cdots\]  violate the rules but \[1=\frac12+\frac13+\frac16\] is a valid EFD.

Now that the rules have been clearly laid out, we can start asking questions about EFDs.

• Will we always eventually have just 1 leftover piece to divide into n equal parts?
• Will we always be able to find a finite list of fractions?
• Are there algorithms for finding an EFD which do not result in denominators dividing each other?
• Is it possible for a given fraction to have more than one EFD?
• What if you are only good at cutting in half so the fractions always have the denominator as a power of 2. Will you always be able to divide the cupcakes fairly?
• What other questions arise this way?