Problem

Source: 2020 Dürer Math Competition Finals E1. 6

Tags: game, combinatorics



(Game) Károly and Dezso wish to count up to $m$ and play the following game in the meantime: they start from $0$ and the two players can add a positive number less than $13$ to the previous number, taking turns. However because of their superstition, if one of them added $x$, then the other one in the next step cannot add $13-x$. Whoever reaches (or surpasses) $m$ first, loses. Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.