This is equivalent to find an integer solution to the following equation: $$27 x + 15y=6$$ This is a standard indeterminate equation which can be solved by Euclidean method. One solution is $(x, y)=\boxed{(4, -2)}$ bcause $$6 = 4\times 15 - 2 \times 27$$ This means: 1) Fill the $15$ liter, and transfer all the water to the $27$ 2) Fill the $15$ again, and use that water to fill the $27$ liter to full. This leaves $3$ liter in the $15$ liter container. 3) Empty the $27$ liter container. 4) Pour the $3$ liter from the $15$ liter container to $27$ liter container. 5) Fill the $15$ liter container and pour all the water to the $27$ liter container. At this point, the later contains $3+15=18$ liter water. 6) Fill the $15$ liter container and then use that water to fill the $27$ liter container to full. This will leave $\boxed{6}$ liter water in the $15$ liter container.