Abstract
Are the problems one encounters at IMOs and the problems one encounters as a research mathematician alike? We will make use of a few examples to show their similarities as well as their differences. The problems chosen come from different areas, but are all related to arrangements of numbers or colors on graphs, and to games one can play with them.
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Recommended Reading
George Pólya, How to Solve It. Princeton University Press, Princeton (1945). This is perhaps the first notable book about problem solving, and it has had a great impact. It remains a timeless classic today.
Paul Halmos, Problems for Mathematicians, Young and Old. The Dolciani Mathematical Expositions. The Mathematical Association of America, Washington (1991). The author has written a few books about problems in research mathematics. This is the most accesible one; it has many problems on the borderline between IMO and research mathematics.
Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning Ways for Your Mathematical Plays, second edition. AK Peters, Wellesley (2004). This (very lively) four-volume book discusses the general theory and also describes many examples of non-random games, played by one or several players. Game of Life (which was invented by one of the authors) is discussed in the last chapter.
Joel L. Schiff, Cellular Automata: A Discrete View of the World. Wiley-Interscience Series in Discrete Mathematics & Optimization. Wiley-Interscience, Hoboken (2008). This is perhaps the best popular introduction to cellular automata, discussing the game of life, the sandpile, and the Ising model, among other things. It is accessible to high-school students while remaining interesting for research mathematicians.
Gregory F. Lawler and Lester N. Coyle, Lectures on Contemporary Probability. Student Mathematical Library, volume 2. The American Mathematical Society, Providence (1999). This is a short collection of lectures in probability, requiring almost no background. It discusses several very modern research topics, from the Self Avoiding Walk to card shuffling.
Alexei L. Efros, Physics and Geometry of Disorder: Percolation Theory. Science for Everyone. Mir, Moscow (1986) This is an introduction to the domain of mathematics that studies phase transitions by considering random colorings of lattices. It is written very nicely and is specifically oriented towards high school students.
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© 2011 Springer-Verlag Berlin Heidelberg
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Smirnov, S. (2011). How do Research Problems Compare with IMO Problems?. In: Schleicher, D., Lackmann, M. (eds) An Invitation to Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19533-4_6
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DOI: https://doi.org/10.1007/978-3-642-19533-4_6
Publisher Name: Springer, Berlin, Heidelberg
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