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Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability inn/ in In n(l + o(l)) balls in it. Suppose instead, that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only in in n/ in 2+0(1) balls -exponentially less than before.doi:10.1145/195058.195412 dblp:conf/stoc/AzarBKU94 fatcat:3dzj3wqs75gwfjbmrgh2ijrfke