Page 23 - Science Focus (Issue 017)

For each line with slope tan(α), it is possible to draw
it such that the first pancake is sliced in half, even for
any value of α. Then we need a function to complete
the argument — here it is most convenient to take the Consider the top piece of bread. First note that
difference in percentages of the area of the second there are three ways you can adjust a knife: translating
pancake as our function, taking the right hand side along the vertical axis (p), and rotating in two angles
of the line, as seen by an observer from the handle — along the plane of the bread’s surface (θ), or the
of the knife, as positive. For example, if 20% of the tilt angle (ϕ). We can reuse the two-dimensional result
pancake lies to the left and 80% of it lies to the right, above by first treating the two pieces of bread as in
our function would equal –20% + 80% = 60%. The sign the case above, and immediately obtain both p and
of the function flips when you rotate α by 180 degrees. θ. Before we proceed, we note that both p and θ
So we can see that there are values of α where the can be described as functions of ϕ — as ϕ changes,
function is negative (more of the pancake is on the p(ϕ) and θ(ϕ) also change accordingly, but both are
left), and where it is positive (more of the pancake is defined above so that they always cut the first two
on the right). solids in half.
Take a pair of values with opposite signs; by The final step is to add the piece of ham. We can
applying IVT, we conclude that there is a certain α then consider f3(ϕ), defined as above as being the
which causes f = 0, indicating that 50% of the pancake difference in percentages of the volume of the piece
lies to the right and 50% lies to the left of the line. Our two of ham, its variable being the remaining tilt angle, ϕ.
pancakes are now sliced into equal halves in one cut! The sign of f3 flips when you rotate ϕ by 180 degrees,
which gives us a pair of positive and negative values
And now it is time to tackle our original problem. we can take as our boundary points. Then, by the IVT,
To simplify things, we’ll split the sandwich into its three we know there is a certain value of ϕ where f3(ϕ) = 0.
parts: two pieces of bread and a piece of ham. Now
the area of the pancake becomes the volume of the Combining the results from all three parts, we then
piece of bread/ham, and the cutting know it is possible that all three pieces are cut into half,
line becomes a 2D plane. which is exactly what we want!

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