Using a simple balance scale, what 4 weights totaling 40 pounds can be used to determine the weight of any object of from 1 to 40 pounds?

(Whole numbers only, no ounces or fractional pounds)

### Solution to the Weights puzzle

The weights are 1, 3, 9, and 27 pounds.

Here's the breakdown, weight by weight, beginning with the heaviest. Note that the 4 weights can be placed on both sides of the scale. The number shown in red is the weight of the actual object being weighed, which is placed on the right side of the scale.

Left Side
Of Scale
Right Side
Of Scale
Total Weight
Each Side
27+9+3+14040
27+9+33939
27+9+338+139
27+9+13737
27+93636
27+935+136
27+9+134+337
27+933+336
27+932+3+136
27+3+13131
27+33030
27+329+130
27+12828
272727
2726+127
27+125+328
2724+327
2723+3+127
27+3+122+931
27+321+930
27+320+9+130
27+119+928
2718+927
2717+9+127
27+116+9+328
2715+9+327
2714+9+3+127
9+3+11313
9+31212
9+311+112
9+11010
999
98+19
9+17+310
96+39
95+3+19
3+144
333
32+13
111

The weights are 1, 3, 9, and 27 pounds - which just happen to be the first 4 powers of 3: 30; 31, 32 and 33.

Actually, this is not so odd (see table above):

2 weights of the first 2 powers of 3 (1+3=4) can be used to weigh all obects from 1 to 4 pounds

3 weights of the first 3 powers of 3 (1+3+9=13) can be used to weigh all obects from 1 to 13 pounds

4 weights of the first 4 powers of 3 (1+3+9+27=40) can be used to weigh all obects from 1 to 40 pounds.

5 weights of the first 5 powers of 3 (1+3+9+27+81=121) can be used to weigh all obects from 1 to 121 pounds

And so on for all other powers of 3.