An ancient balance was found along with 5 weights.

Objects placed on one side of the balance could be counter-balanced by a weight on the other side of the balance to determine the object's weight. Also, the object and a weight(s) could be placed on one side of the balance and counter balanced to determine its weight, if necessary.

By using only these 5 weights, the ancients were able to determine the weight of objects weighting 1, 2, 3, ..., 120, 121 grams. What are the 5 weights that the archeologists found?

The answer might be 1, 3, 9, 27, 81, which are incidentally powers of 3.

A short description of the solution:

1 = 1
2 = 3-1
3 = 3
4 = 3+1
5 = 9-3-1
...
9 = 9
...
14 = 9+3+1
15 = 27-9-3-1
...
27 = 27
...
40 = 27+9+3+1
41 = 81-27-9-3-1
...
81 = 81
...
121 = 81+27+9+3+1

The '...' can be filled by extrapolating the results from the previous sets of weights.

Posted by delvin on 2003-02-13 13:17:06