Weigh It!

In a normal pan balance (as used by, say, a vegetable vendor) the item to be bought is balanced against a known combination of weights. The weights may be applied on either side of the pan--for instance, to weigh 1500 g of onions, the vendor may either place the onions in one pan and a 1000 g and 500 g weight in the other, or place the onions along with a 500 g weight in one pan and a 2000 g weight in the other pan.

We would like to be able to weigh all quantities between 1 g and 4000 g to the nearest gram using such a balance. One way to do this is to have 12 weights weighing 1 g, 2 g, 4 g, 8 g, 16 g,..., 2048 g.

1. Design a smaller set of weights to achieve the same objective.

2. In general, to weigh items between 1 g and n g to the nearest gram, what is the smallest sequence of weights we can use (in terms of n)?

Level:Medium-tough