This afternoon, you received 4 boxes full of nails. They were labeled A, B, C, and D. You were told that two of the labels were switched on accident and that you must find out which ones.
You have several scales with which you can weigh nails against each other. The people here are picky about efficiency, and you'd like to do all the weighings at once, using as few scales as possible. This means you can't change your weighing strategy according to the results of the first weighing.
What is the smallest number of scales needed to figure out which nails were switched if:
- You know that the order from lightest to heaviest is A, B, C, then D, or
- You know that an A nail is 1.9 g, B is 2.0 g, C is 2.1 g, D is 2.2 g.