Water washes away an oily substance that protects your skin. When that happens, a certain type of dead cell on your skin will absorb the water causing the cells to swell up, but the layer underneath does not.
When you are in water for extended periods, this substance gets washed away from your skin and hair, removing this protective layer. So the swelling, combined with the places the skin is connected underneath to the non-swelled layer, causes your finger tips to look all wrinkly.
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