The Ham Sandwich Theorem is a mathematical theorem about bisecting figures which is made easier to understand by stating it in terms we can all relate to. Mathematicians particularly like to state obscure mathematical facts in terms of food. For example, you can read about envy-free cake division, trisecting a Toblerone or the mathematical blancmange. Here's what they have to say about the humble ham sandwich:

Given a ham sandwich made from two pieces of bread and a slice of ham, it is always possible to bisect all three pieces with a single straight cut of the knife.

(Bisecting here means cutting each object into two pieces of equal volume.)

It sounds reasonable at first. Imagine a square slice of bread. On top of that put a square slice of ham, then another square slice of bread. It's obvious you can cut the whole thing down the middle and bisect all three parts of the sandwich. But before you make your cut, move the top slice so that it is crooked, turned with respect to the other two pieces and only half covering the ham. Is it still so obvious that you can bisect all three pieces with a single slice of your knife?

The theorem tells us that it can be done, but doesn't tell us how to do it. The knife cut will more than likely be at an angle, both to the bread and the ham.

The slices of bread and the chunk of ham don't have to be all the same size or shape. They don't have to be lined up. They don't even have to be together - the three sandwich components can be widely separated in space. The theorem tells us that regardless of all these, bisection is still possible with only one cut. Where the pieces are far removed from each other, it is best to image the single cut as a flat plane cutting through all three pieces simultaneously.

The Ham Sandwich Theorem was proposed by a Polish mathematician, Hugo Steinhaus (1887-1972) in 1938. It was proved by another Polish mathematician, Stefan Banach (1892-1945), who did this by showing that it was equivalent to another already proved theorem, the Borsuk-Ulam theorem. This had been formulated by Stanislaw Ulam and proved by Karol Borsuk in 1933.

The Generalised Ham Sandwich Theorem

The Ham Sandwich Theorem can be generalised to any number of dimensions:

Given n measurable objects in n-dimensional space, it is possible to divide all of them in half (with respect to their measure) with a single (n-1)-dimensional hyperplane.

So for example in seven dimensions, it is possible to bisect seven objects (with respect to their 7-d hypervolume) using a single cut of a 6-dimensional hyperplane.

Doesn't have quite the same ring to it as the ham sandwich version, does it?

The Pancake Theorem with a Proof

The Ham Sandwich Theorem is a three-dimensional version of the two-dimensional  Pancake Theorem. This says that given two flat pancakes side by side on a table, they both can be bisected simultaneously (into two pieces of equal area) by a single straight cut.

Because the Pancake Theorem involves only two dimensions it is reasonably simple to prove, although the proof has a lot of steps. It makes use of the concept of a continuous function. Before considering the pancakes, we'll talk about Jack.

Jack Doubles His Height!

Jack is 20 years old. When he was 10, he was only 3 feet tall. Now at 20, he is 6 feet tall (earning him the nickname 'Steeplejack'). Sometimes Jack grew quickly and sometimes he grew slowly, but he grew in a continuous fashion so he passed through all the heights between 3 and 6 feet. We can't say at what age he was 4 feet tall without further information, but we know that he must have been exactly 4 feet high at some age - he never jumped from just under 4 feet to just over 4 feet in an instant.

This is the important thing about continuous change - it must go through all the intermediate values.

Bisecting One Pancake

Let's look at one of our pancakes, which we'll assume is lying flat on a table.

• Pick a direction, say northeast.
• Take a long sword¹ and position it horizontally over the table so that it is pointing northeast and is entirely on one side the pancake. Or we could say all the pancake is on one side of the sword.
• We can now slowly move the sword across the pancake, keeping it pointing northeast, until all the pancake is on the other side of the sword.
• Since the amount of pancake on each side of the sword changes continuously as we move the sword, there must be a position where exactly half (the area of) the pancake is on one side and half is on the other side. At this position, the sword bisects the pancake. Or it would if we let it fall.

So for any random direction, there is guaranteed to be a line which bisects the pancake.

Bisecting Two Pancakes

Now we think about the two pancakes lying on the table side by side.

• We draw a large circle on the table, big enough that both the pancakes are inside it.
• We place a straight rod across the centre of the circle so that it becomes a diameter of the circle.
• We label the two ends of the rod '+' and '-', creating a 'plus' side and a 'minus' side of the rod.
• We draw two lines at right angles to this rod so that each line bisects a pancake. We've already shown that this is always possible.
• If we're really lucky, these two lines will be in fact the same line; by complete chance, we've found the line that bisects both pancakes. Much more likely, we have two parallel lines. Each bisects one of the pancakes and is at 90° to the rod.
• We measure the distance from where the first line crosses the rod to the centre of the circle, calling it p. If the point is on the minus side of the rod, we call this a negative value of p. If it's on the plus side, we call it a positive value of p.
• We measure the distance from where the second line crosses the rod to the centre of the circle, calling it q. Again we label it positive or negative depending on which side of the rod it is on.
• We subtract q from p to give us r which can be positive or negative. It represents the distance between the two bisector lines.
• Now finally we turn the rod slowly through 180° around the centre of the circle. As we turn the rod, the two bisecting lines will move also so that they are still at 90° to the rod, but they will have to move from side to side so that they still bisect their respective pancakes. The distances p and q will vary and r, the difference between them, will also vary.
• We find that over the course of the 180° turn, the values p, q and r all change sign. If p was positive at the start, it will be negative at the end of the turn and vice versa. The same applies to q and since r is p − q, it also applies to r.
• Since r changes sign from positive to negative, or from negative to positive, over the course of our 180° turn, and since it changes continuously over the turn, there must be a position of the rod at which r is 0.
• If r is 0, then the two bisecting lines lie on top of each other and are in fact a single line. So there is a single line which bisects both pancakes.

We've proved the theorem!

The proof of the Ham Sandwich and Generalised Ham Sandwich theorems are more complicated, but use many of the same concepts that we've used here.