Question 114
We have $a + b$ bowls aligned in a row. The first $a$ bowls each contain an apple, while the last $b$ bowls each contain a pear.
An operation consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, where $i$ and $j$ are two integers such that $i-j$ is even (a bowl may contain several fruits).
We want to reach the final situation in which the first $b$ bowls contain a pear and the last $a$ bowls contain an apple. Show that this is possible if and only if $ab$ is even.
An operation consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, where $i$ and $j$ are two integers such that $i-j$ is even (a bowl may contain several fruits).
We want to reach the final situation in which the first $b$ bowls contain a pear and the last $a$ bowls contain an apple. Show that this is possible if and only if $ab$ is even.