We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the others a priori proba
bility of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions that says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the Thue-Morse sequence in terms of certain expansions in fractional bases close to, but greater than, 1.
We consider the following problem arising from the study of human problem solving: Let $G$ be a vertex-weighted graph with marked in and out vertices. Suppose a random walker begins at the in-vertex, steps to neighbors of vertices with probability pr
oportional to their weights, and stops upon reaching the out-vertex. Could one deduce the weights from the paths that many such walkers take? We analyze an iterative numerical solution to this reconstruction problem, in particular, given the empirical mean occupation times of the walkers. In the process, a result concerning the differentiation of a matrix pseudoinverse is given, which may be of independent interest. We then consider the existence of a choice of weights for the given occupation times, formulating a natural conjecture to the effect that -- barring obvious obstructions -- a solution always exists. It is shown that the conjecture holds for a class of graphs that includes all trees and complete graphs. Several open problems are discussed.
