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What does a filter do?
A filter is like a very fine sieve. The fibres in the paper are a mesh that holds back small pieces of undissolved solid, while letting the liquid through.

How can you get solids back after they have dissolved?
You can't get them back by filtering. They will pass through a filter with the liquid. But you can get them back if the liquid evaporates - becomes a gas or vapour and floats off in the air. That's how crystals can be formed as the liquid is lost.

Why is water special?
Water is unique because it actually gets bigger when it freezes. Almost all liquids shrink when they freeze, so that the solid is smaller than the liquid. If you pour a molten liquid like wax or metal into a mould, it will shrink as it cools and not quite fill it. But if you pour water into an ice tray, the ice it makes will be bigger than the tray. This is a good thing, because it makes ice lighter than water - so ice floats, and life can go on underneath it - in ponds and polar seas.