As I was telling you in my previous post (link) I was at Stuttgart, and the time-table was more intense than I had expected at the beginning, though I enjoyed it a lot… It was really interesting, and it will take time to share everything I was doing there, but I promise that I will post as much as possible.
I will start with my first day, obviously. As a normal 1st day, it was more about presenting the university and the city, than math, but we have also done some ‘Mathematics Problem Solving’. The lecture-presentation was made by Prof. Dr. Fischer and it was extremely interesting.
We started with some general ideas about mathematics:
- ‘Mathematics is the art of problem solving.’ (by G. Polya)
- ‘I do believe that problems are the heart of mathematics.’ (by P.Halmos)
- Mathematics is a living, breathing, changing organism with many facets… It is creative, powerful, and even artistic.
- Mathematics is not a set of formulas to be applied to a list of problems at the end of textbook chapters.
- Mathematics is thinking not computing.
- Mathematical problem solving requires creativity (have an idea, see something) and intuition (feeling), but also precise and logical thinking.
Then we discussed in groups some easy math problems; trying to use some of our basic knowledge and not complicated things. Here are some of the problems (I will post the proofs later to give you some thinking time):
- Is √2+√3 rational?
- There is no polynomial p(x) with integer coefficients such that p(1)=4 and p(3)=5 hold.
- Find a formula for the value of the sum 1*1!+2*2!+…+n*n!
- p(x) and q(x) are polynomials with equal coefficients, but in reverse order. Is there any relation between the roots of p and q?
- Is there in every year a Friday the 13th?
Then we did some harder questions in which he was explaining some important concepts that are used a lot in mathematics and also in industry or finance/insurance, such as the pigeonhole principle, divide and conquer or extremum principle. Here are some of the problems I liked the most (again I will post the solutions later to let you digest these harder questions):
- Let there be given 5 lattice points (points with integer coordinates) in the plane. Show that there is a lattice point on the interior of one of the line segments joining 2 of these points. How many points do we need if we consider the 3D space instead of the plane?
- On a certain island we have: 13 green chameleons; 17 red ones and 24 brown. Whenever 2 chameleons of different colors meet, they both change their color into the 3rd color. Is it possible that at one time all chameleons have the same color?
- In the plane a set of 2n points is given, n points are colored red and the remaining n blue. Every red point is connected with a blue point by a straight line segment. Show that it is always possible to do this so that these line segments do not intersect.
- On a circle n points are selected and the chords joining them in pairs are drawn. Assuming that no 3 of these chords are concurrent (except at the endpoints), how many points of intersection are there?
In the end he also recommended some books that are now on my must read list. Here are some:
- How to solve it: A new aspect of mathematical method by G. Polya (1945) [this is the 1st book printed about this subject]
- Mathematical discovery: on understanding, learning and teaching problem solving by G. Polya (1962)
- Problem-solving through problems by L.C.Larson (1983)
- Mathematical thinking: problem-solving and proof by J.P.D’Angelo, D.B.West (1997)
- Problem-solving strategies by A.Engel (1998)
- Principles of mathematical problem solving by M.J.Erickson, J.Flowers (1999)
It was an incredible fun presentation, and I enjoyed the way he was explaining everything to us. He changed the way in which I think about math problems, but also at life problems in general. Because if we think more about this, we use the same thinking procedures to solve our everyday problems. And this kind of lectures really help us understand life in a different way, and help us on our thinking process ( I am jealous that the students there in Stuttgart have this kind of lectures regularly). I totally recommend to check anything related with problem solving (math ones, especially) you will not regret it.