If 100 coins are jammed in, 400 coins can cover the table
A hundred identical circular coins lie flat on a rectangular table without overlapping. A coin counts as "on the table" when its centre is on the table (it may overhang the edge). The coins are packed so tightly that no further coin can be placed on the table without overlapping one of them.
Prove that 400 coins of the same size can be arranged, overlapping as needed, so that every point of the table is covered by at least one coin.
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Why can no more coins be added? Turn that into a statement about how far every point of the table is from the nearest coin centre. Then think about doubling and halving.
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