Problem #PRU-67176

Problem

The king possesses $7$ bags of gold coins, each containing $100$ coins. While the coins in each bag appear identical, they vary in weight and they cannot be told apart by looking. The king recalls that within these bags, one contains coins that weigh $7$g each, another has coins weighing $8$g, the third bag contains coins weighing $9$g, the fourth has coins weighing $10$g, the fifth contains coins weighing $11$g, the sixth holds coins weighing $12$g, and finally, the seventh bag contains coins weighing $13$g each. However, he cannot remember which bag corresponds to which coin weight.
The king reported his situation to his chancellor, pointing to one of the bags, and asked how to determine the weight of the coins in that bag. The chancellor has large two-cup scales without weights. These scales can precisely indicate whether the weights on the cups are equal or, if not, which cup is heavier. Can the chancellor ascertain which coins are in the bag indicated by the king, using no more than two weightings? The chancellor is permitted to take as many coins as necessary to conduct the weightings.