A 40 pound rock drops and breaks into 4 pieces. The weight of all the pieces are whole number of pounds and the 4 pieces can be used to weigh every weight between 1 and 40. What are the weights of the pieces?

To weigh 1 kg, that's obvious.
To weigh 2 kg, put 3 kg on one side, 1 kg on the other. Put your object with the 1 kg weight and make it balance.
To weigh 3 kg, that's obvious
To weigh 4 kg --> 3 kg + 1 kg
To weigh 5 kg --> 9 kg on one side, 3 kg + 1 kg on the other
To weigh 6 kg --> 9 kg vs. 3 kg.
To weigh 7 kg --> 9+1 kg. vs. 3 kg.
To weigh 8 kg --> 9 vs. 1
9 --> 9
10 --> 9 + 1
11 --> 9 + 3 | 1
12 --> 9 + 3
13 --> 9 + 3 + 1
14 --> 27 | 9 + 3 + 1
15 --> 27 | 9 + 3
etc.

You should be able to follow the pattern all the way to 40 kg.