Baron Münchhausen Redeems Himself: Bounds for a Coin-Weighing Puzzle

We investigate a coin-weighing puzzle that appeared in the 1991 Moscow Math Olympiad. We generalize the puzzle by varying the number of participating coins, and deduce an upper bound on the number of weighings needed to solve the puzzle that is noticeably better than the trivial upper bound. In particular, we show that logarithmically-many weighings on a balance suffice.

Linearization of holomorphic germs with quasi-parabolic fixed points

Let f be a germ of a holomorphic diffeomorphism of $\mathbb {C}^n$ with the origin O being a quasi-parabolic fixed point, i.e. the spectrum of dfO consists of 1 and e2iπθj with $\theta _j\in \mathbb {R}\!\setminus \!\mathbb {Q}$. We show that f is locally holomorphically conjugated to its linear part, if f is of some particular form and its eigenvalues satisfy certain arithmetic conditions. When the spectrum of dfO does not consist of any 1’s, this is the classical result of Siegel [C. L. Siegel. Iteration of analytic functions. Ann. of Math.43 (1942), 607–612] and Brjuno [A. D. Brjuno. Analytic form of differential equations. Trans. Moscow Math. Soc.25 (1971), 131–288; 26 (1972), 199–239].