## June 26, 2005

### Unsolved Problems

These are the 24 problems from the wonderful book "Old and new unsolved problems in plane geometry and number theory" by Victor Klee and Stan Wagon, MAA, 1991. Some problems are too cryptic to understand without the book. I'll type more explanations some day...

#### Plane Geometry

• Is each reflecting polygonal region illuminable?
(Imagine a room in a 2-D universe. If the walls form a polygonal shape
and they reflect light, can you always illuminate the whole room with
a single light source?)
• A point P inside a closed convex curve C is called equichordal if every chord of C drawn through point P has the same length. Is there a curve with two equichordal points?
(Marek Rychlik kindly informed me of his solution to this problem.  His homepage has references for the published work.)
• When congruent disks are pushed closer together, can the area of their union increase?
• If a convex body C contains a translate of each plane set of unit diameter, how small can C's area be?
• What is the smallest number f(n) such that any set of f(n) points (non-collinear) always contains the vertices of some convex n-gon?
• If n points are not collinear, must one of them lie on at least n/3 connecting lines?
• Is there a polygon that tiles the plane but cannot do so periodically?
• What is the minimum number of colors for painting the plane so that no two points at unit distance receive the same color?
• Can a circle be decomposed into finitely many sets that can be rearranged to form a square?
• Does the plane contain a dense rational set?
• Does every simple closed curve in the plane contain all four vertices of some square?
• Does each nonseparating plane continuum have the fixed-point property?

#### Number Theory

• Do there exist positive integers x, y, and z and an integer n>=3 such that x^n + y^n = z^n?
(Too late for this one :^) Wiles got it in 1993.  Check out http://www.best.com/~cgd/home/flt/flt01.htm for the proof.
• Does there exist a box with integer sides such that the three face diagonals and the main diagonal all have integer lengths?
• Does the greedy algorithm always succeed in expressing a fraction with odd denominator as a sum of unit fractions with odd denominator?
• Is there an odd perfect number? Are there infinitely many even perfect numbers?
(A perfect number is one whose factors add up to itself, e.g. 6, 28)
• Do the nontrivial zeros of the Riemann zeta function all have real part 1/2?
• Is there a polynomial-time algorithm for obtaining a number's prime factorization?
• Is every positive integer eventually taken to the value 1 by the 3n+1 function?
(3n+1 function is defined as f(n) = n/2 (if n even); 3n+1 (if n odd))

• Is there an algorithm to decide whether a polynomial with integer coefficients has a rational root?

#### Interesting Real Numbers

• Are the digits in the decimal expansion of pi devoid of any pattern?
• Are pi and e algebraically independent? Is their ratio rational?
• If an irrational number is real-time computable, is it necessarily transcendental? Is sqrt(2) real-time computable?
(A number is real-time computable if the amount of time spent computing its digits is linear in the number of digits.)
• Is 1 + 1/2^5 + 1/3^5 + ... irrational?