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The theory of pizza

Have you ever thought about using theories of geometry to cut a round pizza for two people to get two pieces weighing the same amount?

Yet someone has thought of it! Quite probably at a dinner among friends, someone realized someone was getting more and someone was getting less, we don't know for sure, but we do know that two scientists, Mabry & Deiermann, have applied a rather simple theory to resolve the issue.

Let's suppose two people want to share the pizza “artistically”: one slice each, taking turns.
If the pizza is divided by making three cuts across its diameter, at least one of which goes through the centre, the two will eat an equal amount of pizza (as shown in the picture to the left).

But what happens if the person cutting the pizza does not go through the centre?
If an even number of cuts is made (for example, 4) they will get the same amount of pizza. We know this thanks to a certain Mr Upton who in 1968 stated that “the sum of the areas of odd segments is equal to the sum of the areas of even segments.”
But what happens if we cut through the pizza 3, 7, 11, or 15 times, or 5, 9, 13, or 17 times?
Everything gets complicated: in the first scenario, the person who gets the slice with the centre of the pizza in it gets more, in the second, the person who gets the slice not containing the centre gets more.

The last two images show how our two scientists used sub segments to explain the solution, and someone has even come up with a hypothesis on the distribution of toppings.

But then we had an idea: couldn't the two people eat a pizza each?

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