If you had 6 quarts in the large pail, it would hold 3 quarts more. So, if it were full, you could pour 3 quarts into a small pail that contains 1 quart (the antecedent state) and you would have the desired 6 quarts.

Fill the large pail. Fill the small pail from it and dump this water. Repeating the process leaves 1 quart in the large pail. Pour that quart into the small pail. Fill the large pail and pour as much as you can into the small pail, leaving exactly 6 quarts in the large pail. Problem solved (Polya, 226-230).