Exploring Mathematical History through Problem Solving

  • Instructor: Péter Juhász
  • Contact: juhasz.peter@renyi.hu
  • Prerequisites: None
  • Text: your class notes

Note: this course is classified both as a math and a culture course.

Course description:

In the second millennium, mathematics underwent tremendous development, and this period is filled with exciting moments when progress was by no means guaranteed. It is, of course, very difficult to know exactly what was going through the minds of mathematicians at the time, but we will try to recall some of these situations and explore the associated problems and their solutions.

This course focuses on the last 800 years of European history of math, with a particular emphasis on Hungarian aspects from the past 200 years. The number of significant Hungarian mathematicians is larger than what the country's population would justify. This is likely due to the tradition of Hungarian mathematics education, especially of talent nurturing. The course also discusses the last 130 years of Hungarian mathematics education, including notable math teachers (Dániel Arany, László Rátz, George Pólya, Tamás Varga, Lajos Pósa), the KöMaL journal, and the wide range of math competitions.

The course consists partly of shorter and longer lectures and related discussions of the above topics, while the other - more significant - part involves thinking through related problems. Following the problem-solving sections, we will discuss in detail the different approaches, solutions, and their historical significance in mathematics.

The course material is influenced by students' interests, but the following topics are highly likely to appear in the content:

  • sum of series (Grandi, Cauchy)
  • infinity: Galileo and Cantor
  • concept of function (Euler, Dirichlet, Cauchy)
  • history of solving cubic equations
  • Hungarian mathematicians and their results:
    • Bolyai--Gerwien-theorem
    • Erdős’ favorite problems
    • Turán theorem
    • The Anonymous group and the Happy end problem
  • Hungarian math education:
    • Arany Dániel, Rátz László and KöMaL math competitions. Students can try out Hungarian competitions like Zrínyi, Bolyai, Dürer, OKTV and Kürschák.
    • special math classes
    • the discovery learning approach: Pólya, Varga, Pósa methods
  • Hungarian specialities:
    • Rubik's cube
    • Gömböc
    • Barkochba (21 questions game)