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The answer is basically friction. A dry product made of wood fibers like tissue or paper towel is very lightweight and when resting against a wooden table has less effective contact and friction between the two surfaces. If the table is tilted, the paper product will slide due to the low friction. A wet paper towel or tissue, however, has absorbed many times its weight in water and has greatly increased in mass. This gives the material more weight and therefore it has more effective contact with the wood table and as such a higher friction coefficient. As a result, when the table is tilted the material will stay in place more effectively.
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