You are setting up a weighing service. Customers will walk in to your shop with an object and you will weigh it for them. We can assume that the objects brought in always have positive weight, and that in fact this weight is an exact integer in pounds. You have just put up a sign on your door: Will weigh anything up to and including N pounds! and now you are about to order the little weights used to counterbalance the customer's object.
Puzzle 1: Assuming that N is 15, what is the smallest number of weights you can order, and still be able to weigh any object up to N pounds?
Puzzle 2: Assuming that N is 2k-1 for some positive integer k, what is the smallest number of weights you can order, and still be able to weigh any object up to N pounds?
Puzzle 3: For each possible value of N from 1 to 15, what is the smallest number of weights?
Puzzle 4: Assume you can put your weights on either side of the scale. For instance, if you have a 4 pound weight and a 10 pound weight, then you could detect a 6 pound object because you put the 4 pound weight on the same dish as the object, and these balance the 10 pound weight. Assuming that N is 13, what is the smallest number of weights you can order?
Puzzle 5: Do puzzle #4, but for any and every (positive) value of N.
I give up, show me the solution.