Question 53
In a table, there are $2009$ coins each with one side white and the other side black. At the beginning all coins are aligned in a row with all their white sides up except one with its black side up. In each step we choose a coin with black side up and flip the two adjacent coins. If we choose the outermost coin, then we flip just one adjacent coin. Find all positions of the black coin at the beginning from which we can reach the state where all coins have black sides up.