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+1 | 40 | 40 |
| 27+9+3 | 39 | 39 |
| 27+9+3 | 38+1 | 39 |
| 27+9+1 | 37 | 37 |
| 27+9 | 36 | 36 |
| 27+9 | 35+1 | 36 |
| 27+9+1 | 34+3 | 37 |
| 27+9 | 33+3 | 36 |
| 27+9 | 32+3+1 | 36 |
| 27+3+1 | 31 | 31 |
| 27+3 | 30 | 30 |
| 27+3 | 29+1 | 30 |
| 27+1 | 28 | 28 |
| 27 | 27 | 27 |
| 27 | 26+1 | 27 |
| 27+1 | 25+3 | 28 |
| 27 | 24+3 | 27 |
| 27 | 23+3+1 | 27 |
| 27+3+1 | 22+9 | 31 |
| 27+3 | 21+9 | 30 |
| 27+3 | 20+9+1 | 30 |
| 27+1 | 19+9 | 28 |
| 27 | 18+9 | 27 |
| 27 | 17+9+1 | 27 |
| 27+1 | 16+9+3 | 28 |
| 27 | 15+9+3 | 27 |
| 27 | 14+9+3+1 | 27 |
| 9+3+1 | 13 | 13 |
| 9+3 | 12 | 12 |
| 9+3 | 11+1 | 12 |
| 9+1 | 10 | 10 |
| 9 | 9 | 9 |
| 9 | 8+1 | 9 |
| 9+1 | 7+3 | 10 |
| 9 | 6+3 | 9 |
| 9 | 5+3+1 | 9 |
| 3+1 | 4 | 4 |
| 3 | 3 | 3 |
| 3 | 2+1 | 3 |
| 1 | 1 | 1 |
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.