It's probably something that doesn't concern the Italians, pizza shouldn't be shared; at most, it can only be split between two people, with your significant other. However, as they have learned from TV series and films, the English and us Americans have this custom of taking a pizza and dividing it among five, six, seven people… For this reason, it doesn't seem so crazy – let's say – that a certain LJ Upton created a theorem that explains how to divide a pizza exactly, so that everyone eats the same amount.
Eight Slices and a Secret Point
It's called the Pizza Theorem, published in Mathematics Magazine in 1968, and it proposes a way to cut that disk of dough exactly the same for each diner. So, you'd think someone had simplified our work, but when mathematics gets involved, as we know, nothing is ever easy.
In any case, according to the theorem, you start by choosing a random point on the pizza and making the first straight cut. You then proceed by making another cut, perpendicular to the first, passing through exactly the same point as the first, thus forming four 90-degree angles. To visualize this better, consider that the slices all have their tips in the same point, regardless of their length. Finally, you cut the four corners obtained in half, always passing through the central point chosen at the beginning, thus creating eight slices. Once cut, the way in which you eat it also follows a precise order: you take any first slice and proceed in an alternating manner, so 1, 3, 5, 7 and the other diner will take slices number 2, 4, 6, 8. This is because, according to the pizza theorem, "the total area of the slices of one color is equal to the area of the slices of the other color." The reason is a question of symmetry: it's true that the cuts don't pass through the center, and it's true that, to the naked eye, the slices aren't equal, but the symmetry of the cuts causes the areas to compensate. So, if we add the alternating slices, the differences in size between a large and a small slice cancel each other out.
The theorem has been explained many times, and Larry Carter and Stan Wagon's "wordless proof" remains one of the most effective. They demonstrate visually and immediately how, by further dividing the segments, each portion finds its exact counterpart among the segments of the other color: in this way, the areas balance each other perfectly, almost like in a puzzle.
And What if There Are More Than Two Diners?
Since eight slices were obtained in the manner explained above, it is clear that the division is valid only if there are an even number of diners. If, however, there are 3 diners, the right angles must be further divided into 3, so as to obtain 12 slices and, therefore, 4 each, to be taken alternating between diners 1, 2, and then 3. Also in this case, the sum of the areas of the slices of the same color is equal to that of the slices of the other colors and, therefore, each diner will have the same amount of pizza.
The "Limits" Of the Theorem and The Conditions to Be Respected
There is a necessary condition for everything we have explained to happen, theorists explain: the number of cuts must be even and a multiple of 4 (this generates 8, 12, 16 slices, and so on). When the number of cuts is odd, such as 3 cuts (6 slices) or 5 cuts (10 slices)—or in the only even exception where there are 2 cuts, which would give 4 slices— the equilibrium is broken. The reason, in this case too, is linked to the symmetry of the circle: only with a sufficient number of cuts does each slice find its "companion" that balances it. If there are six or ten slices, this mathematical correspondence no longer exists.
When the number of slices is not even and a multiple of four, the property demonstrated by the Pizza Theorem no longer holds generally. In these cases, the sum of the areas of alternating slices is not guaranteed to be exactly equal. With an odd number of cuts passing through the same point inside the disc, the rotational symmetry on which the theorem is based breaks down, and perfect compensation between the slices does not automatically occur.
Subsequent studies analyzed these configurations, showing that the difference between the slices depends on the angular arrangement of the cuts. In some configurations, the slice encompassing the center of the disc may be slightly larger; in others, it may be slightly smaller. It's not the "center" itself that determines the advantage, but how the areas of the sectors are distributed around the chosen point.
In short: only when the cuts are an even number and a multiple of four is the equality of the alternating sums guaranteed for any chosen interior point. In other cases, small differences in area may arise, depending on the specific geometry of the subdivision.
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