Unsolved Problems: Part II

This is a continuation of my post from last Thursday on simple unsolved problems in mathematics.

Inscribed Square Problem

The inscribed square problem is an unsolved problem in geometry, proposed by German mathematician Otto Toeplitz in 1911. It asks whether it is possible to draw a square inside a closed loop so that all four corners of the square are contained in this closed curve.

Some early positive results were obtained by Arnold Emch and Lev Schnirelmann, but the general case still remains open. An example of a positive result is shown below:

This has already been solved for other shapes, such as triangle and rectangles, but the case with squares is proving trickier to solve.

Happy Ending Problem

The Happy Ending problem, so named by Paul Erdős because it led to the marriage of Esther Klein and George Szekeres, is the following statement:

“Theorem: any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.”

In other words, if you make five dots on a piece of paper, you should always be able to connect four of them to create a convex quadrilateral (i.e. a shape with four sides where all the corners are less than 180 degrees) regardless of the position of these dots.

Although the result for a four sided shape is known (and in fact it is known for a pentagon – 9 dots are needed – and a hexagon – 17 dots are needed), it is still unknown how many dots is needed to create a shape with a greater number of sides.

Mathematicians speculate that the equation which determines the number of dots M can be related to the number of sides in the shape N by the following equation:

M=1+2N-2

However, it has only yet been proved that this will give a number AT LEAST as big as the answer.

Hope you enjoyed this mini-series! Let me know if you want me to continue it. M x