turkmath.org

The solitary game `Lights out` is discussed in the work of Sutner, Goldwasser- Klostermayer-Ward, Barua-Sarkar, Hunziker-Machiavello-Park, et al. Suppose that the interruptors in a building are synchronized so that switching o in one oce automatically changes the state to the opposite in all neighbor oces separated by a wall. How can one switch all the lights off? Numerous problems arose and mainly remain open. They can be formulated as spectral problems, in terms of harmonic analysis, or as problems about Chebychev polynomials. The Fourier transform translates these problems into unusual but intriguing questions in arithmetic algebraic geometry, concerning torsion on a certain elliptic curve. Some advances, generalizations, and computer simulations done by Don Zagier will be presented.