Sandwich problem ⁠

Can you cut by a planar section a sand­wich consisting of a slice of bread, a slice of cheese and a slice of ham in such a way that both sides contain the same amounts of bread, cheese and ham? We will show that you can...

Consider first the two-dimen­sional problem. Suppose we are given any two regions in the plane. Does id exists a straight line that simul­ta­ne­ously bisects the first region and the second one in two parts with exactly the same area?

To demon­strate that it exists, let’s initially consider only one of the regions. We choose an arbi­trary direc­tion. Is there a line with this direc­tion that divides the region into two parts of equal area? We show that this line exists any direc­tion is chosen. Take the straight line with the chosen direc­tion far from the region so that the region is completely on the same side with respect to it, for instance on the right side. We will build two graphs, repre­senting the areas of the region that are located on the right and on the left of the line, as func­tions of the distance of the straight line from its initial posi­tion. At first the whole region is on the right of the line, which means that the left element of the histogram is zero whereas the right element repre­sents the entire area of the region. We start to move the line to the right so that it is always parallel to its initial posi­tion. During this shift the area of the region located to the right of the line will decrease contin­u­ously, while the area to the left will increase contin­u­ously. At the end, the whole region will be on the left of the line. The left part of the histogram (in blue) then repre­sents the area of the entire region, and the right one will be zero.

If we look at the graphs repre­senting the areas at left and at right of the line as func­tions on the distance of the line from its initial posi­tion, we will see that they have some­where a point of inter­sec­tion, precisely because of their conti­nuity. This point tell us exactly the loca­tion of the line that divides the area of the region into two equal parts.

Since the direc­tion of the line was chosen arbi­trarily, a line that divides the region into two equal parts exists in any direc­tion.

Let us come back to the case of two regions. We will consider only those straight lines that cut the first region (on the left) into two equal parts, and we start choosing the line that is hori­zontal and points to the right. It will cut the second region (on the right) in some way. Now we change the direc­tion of the line, remaining inside the set of straight lines that bisect the first region. We construct the graph of the differ­ence of areas of the first region located at the left and at the right of the line, depending on the angle made by the line with the initial one, which is hori­zontal. Initially this differ­ence is nega­tive. Contin­u­ously changing the direc­tion of the line we will return to the starting posi­tion, but inverted, so that the areas at left and at right are exchanged. At this moment the value of the differ­ence will be posi­tive. Since the differ­ence changes contin­u­ously, its graph, going from a nega­tive to a posi­tive value must meet the line of the zero value. This value corre­sponds to an angle that deter­mines the line that divides the second region into two parts of equal areas. In this way we have found the straight line which divides both regions into two parts of equal areas.

Here is how the Bolzano-Cauchy theorem is used. Unfor­tu­nately, how to draw this line when we are given two regions of arbi­trary forms, arbi­trarily arranged in the plane, we cannot know without using other ideas and knowl­edge. But it exists, for any pair of regions! The theo­rems of this type are called “theo­rems of exis­tence.”

Let us now return to the three-dimen­sional case. Instead of two regions in the two-dimen­sional plane we will consider three arbi­trary objects, arbi­trarily arranged in the space. Instead of the areas, we will consider the volumes, and instead of a straight line, a plane. It happens that in this case, by an argu­ment similar to that used in the planar case, we can prove a theorem of exis­tence. For any three objects there is a plane that divides simul­ta­ne­ously each of them into two parts of equal volume.

In order life have more flavour, consider a sand­wich made of bread, cheese and ham. They are three objects arranged in some way with respect to each other. Try to show that there is a plane that cuts the ham into equal parts and simul­ta­ne­ously cuts the cheese and the bread in equal parts. Using addi­tional argu­ments, this plane was found for the sand­wich shown in the film, and indeed it is evident from the veri­fi­ca­tion that all three objects were divided into two equal parts!