How do we do a divide a pizza so that each diner eats exactly the same quantity? A fairly easy solution to put into practice gives the Pizza theorem. It is a theorem, born from a challenge published in the magazine Mathematics Magazine In 1967, according to which it is enough to divide the pizza with straight cuts, which everyone passes through the same point following some simple rules. Let’s see what these rules are and how the theorem helps us to divide the pizza equally according to the number of diners.
Let’s start by the case of only 2 people:
- First you make a straight cut at random that divides the pizza into two parts
- Then you make another cut, perpendicular to the first, in order to form four different wedges but which create 4 angles of 90 ° in a selected point at random
- We conclude with two cuts that divide all the corners that have formed in point 2
At this point Just distribute the 8 wedges in an alternating way Since, according to the pizza theorem
The total of the area of the wedges of a color is the same as the area of the wedges of the other color (see figure above)
The theorem was demonstrated in various ways among which, for immediacy, the “stands outDemonstration without words“developed by Larry Carter and Stan Wagon in 1994 (see figure below) in which each segment of a color is divided into portions that each have their own counterparty, identical, among the wedges of the other color.
But what does this method have convenient compared to dividing the pizza into two with a single cut? If we want to divide the pizza in an approximate way it is certainly more practical to give a cut in the middle by eye, but if we want to divide the pizza exactly things change: a cut that divides exactly halfway a pizza must necessarily pass exactly through the center, but finding the precise center of a pizza is very difficult. With the method of Pizza theoremon the other hand, the cuts can meet in a random point of the pizza and there is no need to find the center!
And if we have More than 2 diners? No problem, Just change point 3 as follows:
We make a number of cuts equal to the number of diners in order to divide any of the 4 wedges of point 2 equally
For example, If the diners are 3 diners must divide the corners into 3 straight obtained with the first two cuts obtaining 12 total wedges to be distributed in an alternating way, 4 each, as shown by the colors of the figure below.
Also in this case the pizza theorem guarantees that the sums of the areas of the wedges of the same color are equivalent, so the amount of pizza will be the same for each diner. However, there remains a doubt: are we sure that the amount of crust and seasoning is divided equally? The answer is yes, the pizza theorem takes us away from this problem: it has been shown, in fact, that with this method of division of the pizza, the edge and seasoning are also distributed equally.
