CTK Insights

17 Oct

A pizza with a hole

The editorial in the Crux Mathematicorum (43(8), October 2017) posed an interesting problem; how to equally share a pizza with a hole.
Editorial from Crux: pizza with a hole

To make the problem solvable, we need to assume a degree of abstraction. For example, if the hole makes it more difficult to divide a pizza, the assumption that it is possible to divide a a pizza without a hole into two equal parts needs to be accepted for a fact.

A simplified problem

Now, to start with a simpler problem, how can we divided a rectangular pizza into equal parts with a single straight cut? We may find the midpoints of the opposite sides and join them with a cut.

There would be even less trouble dividing a circular pizza: any cut through the center of the pizza could be considered a fair division. Why so? Because circle has central symmetry such that the two halves of a circle fall on top of each other when the tray holding the pizza is rotated 180^{\circ} around the pizza's center. But then rectangle also has a center and any line through its center divides rectangle into equal parts - rectangle is also centrally symmetric.

The real problem

Now we are in a position to tackle the problem from the Crux. Any line through the center of the rectangular pizza divides it into equal parts. One of these lines stands out. It's the one that divides into equal parts the circular hole, that's the line that passes through the hole's center. Thus to solve the problem we cut through the centers of the rectangle and that of the hole.

But that's not the only solution. In fact there are infinitely many more, although to find any of these in a constructive manner is rather difficult, if not impossible.

Draw a straight line in any direction outside the pizza. Now begin moving it towards the pizza perpendicular to its direction. The line will cross the pizza eventually. First the "front or remaining" portion of the pizza will be greater than the one already passed over. As the line proceeds with its movement, the former part decreases whereas the latter increases. With an appeal to continuity, we may conclude that at some point the two parts had the same worth. The key here is what is known as Bolzano's or the Intermediate Value Theorem. There are ways and ways to employ the theorem, e.g., by letting the line rotate around a fixed point.

The "Ham sandwich theorem" is the most famous statement of this kind, see, e.g., tippingpoint video on youTube.

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