Question 35
Max has $2015$ jars labelled with the numbers $1$ to $2015$ and an unlimited supply of coins. Consider the following starting configurations:
a. All jars are empty.
b. Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins.
c. Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin.
Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ coins to each jar except to the jar $n$.
Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar.
(Birgit Vera Schmidt)
a. All jars are empty.
b. Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins.
c. Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin.
Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ coins to each jar except to the jar $n$.
Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar.
(Birgit Vera Schmidt)