What are coffin problems in mathematics?

“Coffin problems” are extremely difficult math problems which, nevertheless, have elementary solutions. They were devised for and used by admission committees at Soviet universities in the 70’s and 80’s to keep Jewish candidates (and other “undesirables”) out of the most prestigious universities, such as Moscow State. Only undesirable candidates were asked these questions in the oral entrance exams.

For more on the history of the practice, see:

• Chapter 3 of Edward Frenkel’s “Love and Math”, or Frenkel’s article “The Fifth Problem: Math and anti-Semitism in the Soviet Union”.
• Alexander Shen’s “Entrance Examinations to the Mekh-Mat”, The Mathematical Intelligencer Vol. 16, No. 4, 1994.
• Tanya Khovanova’s “Coffins” page, and her article "Killer Problems" in The American Mathematical Monthly, Vol. 119, No. 10 (2012), pp. 815-82.
• Articles by Mark and Anatoly Vershik.
• Another reference suggested by Ted Alper is the book “You Failed your Math Test, Comrade Einstein” by M. Shifman.

The articles by Shen and Khovanova, and the solution set by Ilan Vardi, contain a large collection of Coffin problems. Some of these problems have also appeared elsewhere, for example in Peter Winkler’s wonderful collection or in various math olympiads.

The tragedy of these problems is that, despite their ignominious origin, many of them are absolute gems, a testimony to the ingenuity and creativity of those tasked with finding problems that are impossible to solve in the setting of an oral exam, while being “defensible” in the sense that short, ingenious, elementary solutions do exist.

In my mind, one of the best of these problems is the following geometry pearl, which is also the one in Winkler’s book (p. 47, “Sphere and Quadrilateral”):

A quadrilateral in space has all of its edges tangent to a sphere. Prove that the points of tangency are coplanar.

(A quadrilateral in space is simply four points A,B,C,D and four straight line segments AB,BC,CD,DA. The four points may or may not lie on a plane.)

Approaching this problem in any of the usual or not-quite-usual ways – with analytic geometry in space, or vector calculus, or complex coordinates on the Riemann sphere – is extremely difficult.

However, an ingenious trick allows for a solution in a few lines. Each vertex of the quadrilateral has two edges emanating from it and tangent to the sphere at two points; the distances d from the vertex to these two points are the same (why?). Put a mass of 1/d at that vertex. For any two vertices, their center of mass is now precisely the point of tangency between them (observe how this doesn’t work with mass of d: you need a small mass for the larger distance!). Therefore, the center of mass of the entire contraption lies at a point which also lies on lines connecting opposite pairs of tangency points, which means the whole thing is in one plane. \blacksquare

Ilan Vardi presents here several alternate solutions, including an algebraic one spanning four pages of calculations. This is not an easy problem by any stretch of the imagination, and the fact that it nevertheless has such a short solution is truly miraculous.