Page 21 - Science Focus (Issue 017)

By Sonia Choy 蔡蒨珩
Hungry Mathematicians and

the Ham Sandwich

肚餓的數學家與他們的火腿三文治

Mathematicians like their food values f(a) and f(b). Basically, the Intermediate Value
(a lot). There are theorems about numbers, functions, Theorem tells you that there is always a number c
circles, triangles — and theorems about pizzas, between a and b (on the x-axis) that produces a
potatoes, cakes, pies, donuts and sandwiches. We’ll number between f(a) and f(b) (on the y-axis) [2] .
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visit one of these here — the Ham Sandwich Theorem, IVT looks intuitive from a graphical point of view
a problem on sandwich-cutting proposed in the late — think of the curve as a mountain. If we start at sea
1930s [1]. level, at the foot of a 200m hill and climb up to the top,
There’s an ongoing debate whether cutting at some point we are bound to pass a point that is at
a sandwich into triangles or rectangles is the right 100m above sea level. We will not go into the details of
method; however, we are considering something a the intimidating math here, but this is the general idea
little less aesthetically pleasing — what if your little and what we need for this argument.
brother has nipped off a corner of your sandwich, One thing to note though is that IVT requires the
or the piece of ham inside has been folded into function to be continuous, which means that it
some odd shape? No matter what the shape of the doesn’t break open in the middle. You can see this
sandwich is, can you still divide the sandwich into in Figure 2; you can never find a number x for which
two equal halves, to be shared amongst two hungry g(x) is between 0 and 1, since the function has a
people? sudden jump in the middle.

Our Tool: The Intermediate Value Theorem The Solution
It might sound absurd, but mathematics tells So what does it have to do with cutting our ham
us that no matter how irregular the shape of the sandwich into two equal halves? It turns out that all we
sandwich is, there is always a way to cut the sandwich have to do to solve this problem is to apply IVT.
into two equal halves in one cut. This is due to a result
in calculus called the Intermediate Value Theorem More conveniently, we will first consider the two-
(IVT). Now that I’ve said the C-word, however, don’t dimensional case, which is a problem of slicing two
flat pancakes into two equal halves. You may have
run away just yet — Figure 1 can help you visualize it.
learned already that a line can be described by its
We will only look at the part of the graph that slope, tan(α), where α is the angle it makes with the
is between a and b, which generates the function positive x-axis, and its y-intercept, c.

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