This puzzle re-uses the same glass vase from Puzzle of the Week #26. I added an encoded solution in the comments of that question.
I have a glass vase, with a square base and vertical sides. The vase is 50mm x 50mm, and 260mm high. The bottom 10mm is solid glass, and the remaining 250mm has 1mm thick glass walls.
The density of glass is 2.4 times that of water. When I pour a small amount of water into the vase, the centre of gravity is brought down slightly. As more water is added, at a certain point the centre of gravity will begin to creep back up.
How much water added to the vase, to the closest millilitre, will give the lowest centre of gravity for the vase and its contents?