Question 7
In a game, Mario moves forward by jumps of length $1$ metre, $a$ metres or $b$ metres, where $a,b$ are positive integers with $1\lt a\lt b$. Given a route of length $m$ metres, where $m$ is a positive integer, Mario always uses this strategy: first take as many jumps of length $b$ as possible without passing the destination; if the destination has not been reached, take as many jumps of length $a$ as possible without passing it; finally take jumps of length $1$. Mario wins if there is no way of reaching the destination in fewer jumps than this strategy. Otherwise Mario loses.
(a) Show that for $(a,b)=(6,14)$, there exists an $m$ for which Mario loses.
(b) Find the number of pairs $(a,b)$, with $1\lt a\lt b\le100$, for which Mario always wins with this strategy for every positive integer $m$.
(a) Show that for $(a,b)=(6,14)$, there exists an $m$ for which Mario loses.
(b) Find the number of pairs $(a,b)$, with $1\lt a\lt b\le100$, for which Mario always wins with this strategy for every positive integer $m$.