Combinatorics · Graph colouring · Parity · Bipartite graphs · Walks
A spider has spun the web shown below: five regular octagons nested one inside the other, with each vertex of an octagon joined by a thread to the corresponding vertex of the neighbouring octagons. The web has \(40\) vertices, and the spider sits on the vertex of the outermost octagon marked in the figure.
It waits until exactly one fly is caught at each of the other \(39\) vertices, and then sets off on a peculiar walk. Each move takes it along one thread to an adjacent vertex. It passes \(25\) vertices and, at the \(26\)th vertex it reaches, eats the fly sitting there if that fly is still there; then it passes another \(25\) vertices and again eats the fly at the \(26\)th if it is still there; and so on for as long as it likes.
At most how many flies can the spider eat?
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2006, 1. letnik, category A, problem 4. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source