Tokarev S.i.

Among 2000 indistinguishable balls, half are aluminum with a mass of 10 g, and the rest are duralumin with a mass of 9.9 g. It is required to separate the balls into two piles such that the masses of the piles are different, but the number of balls in them is the same. What is the smallest number of weighings on a balance scale without weights that can achieve this?

#
Let's compare the mass of 667 balls with the mass of another 667 balls. If the masses of these two piles are not equal, the required condition is met.

Suppose the specified masses are equal. Then the mass of 666 balls that did not participate in the weighing is not equal to the mass of any 666 balls lying on one of the pans of the scales.

Indeed, if each of the weighed piles contains exactly $k$ aluminum balls, then among any 666 balls of any of these piles, the number of aluminum balls is $k$ or $k-1$. Meanwhile, among the 666 balls that did not participate in the weighing, there are exactly $1000-2k$ aluminum balls. It remains to note that neither the equality $k = 1000 - 2k$, nor the equality $k - 1 = 1000 - 2k$ can be satisfied for any integer $k$.

## Answer

In one weighing.