MathDB

There are

a+b

bowls arranged in a row, numbered

1

through

a+b

, where

a

and

b

are given positive integers. Initially, each of the first

a

bowls contains an apple, and each of the last

b

bowls contains a pear. A legal move consists of moving an apple from bowl

i

to bowl

i+1

and a pear from bowl

j

to bowl

j-1

, provided that the difference

i-j

is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first

b

bowls each containing a pear and the last

a

bowls each containing an apple. Show that this is possible if and only if the product

ab

is even.

Problem Statement