# Cutting a cube with a plane

Me and a couple of friends have recently started solving Math problems in our breaks at school. We came along the Georg Mohr tournament, which is Denmark's first qualifications tournament for the IMO.

Thinking about joining this year, we took a look at the problems of the previous years, and came about the following problem:

The plane shape that appears, when you cut a cube with a plane, can never be:

A pentagon, a line segment, a rectangle (that is not quadratic), a right angled triangle or an equilateral triangle.

We took a shot at it, and concluded that:

• A pentagon is possible (Cutting diagonally on a face away from the center of said face).
• A line segment is possible (Cutting as close to the bottom of a face as possible)
• A rectangle is possible (Cutting a straight line through the center of a face leaves a triangle on both sides)
• A right angled triangle is possible (Cut diagonally through the center of a face)

After finishing the rest of the problems, we looked at the answer sheet, stating that the correct answer was a right angled triangle.

We went to Google to try to figure out why, and we found out how to produce the equilateral triangle, but are now left with the question of how the right angled triangle is not possible?

We think that we're interpreting the question wrong, so for any danes that wanna check our translation and/or interpretation, the original question is written as such:

Den plane figur, der fremkommer, når man skærer en terning med en plan, kan aldrig være:

So if I didn't make it clear, the question is why is it impossible to produce a right angled triangle in this way?

I think I understand your point of confusion. What you considered seems to be the shape of the face after the cut. That would be the same as just cutting a square with a line.

What the questions asks for, is the shape of the two new internal faces that appears after the cut. For example, if you cut along an edge to the opposite edge, you would get a rectangle of dimensions $$1\times\sqrt2$$. It is the intersection of the cube and the plane.

I speak Danish by the way.

This illustration shows what they mean. (Found with google on http://cococubed.asu.edu/code_pages/raybox.shtml).

• I thought it would be a quite simple answer indeed, thanks a lot! – ItzBenteThePig Nov 9 '19 at 14:13

A plane through the diagonal of one face and an opposite vertex produces either an equilateral triangle or a rectangle.

The rectangle does contain right angled triangles - is that what you were thinking?