ou have only four weights and a balance scale. If your task is to weigh ANY load from 1 to 40 kg (integers only) then how much should each of your 4 weights weigh?
This a tough question. I found the answer with an online search,
at http://www.funtrivia.com/askft/Question26870.html ,
but probably would not have been able to figure it out myself for hours.
You do it with four weights (1, 3, 9 and 27) by putting weights on both sides of the scale. For example, to weigh something that is 25 pounds, you put 27 and 1 on one side, 3 on the other side, and the item you are weighing on the same side as the 3.
For an 11 lb weight, put 9+3 lb on one side and 1 lb with the unknown on the other side. For 19, put 27+1 on one side and unknown + 9 on the other side. There is always a way.
Note that the total of all weights is 40, so can measure a 40 lb weight by putting them all on one side.
To solve this problem, let's start by considering the available weights. As you mentioned, we have only four weights.
To weigh any load from 1 to 40 kg, we can assign weights to be powers of 3. This is because all numbers from 1 to 40 can be represented as sums of distinct powers of 3.
Let's see how this system works:
1 = 3^0 -> We need the weight of the smallest weight for 1 kg.
3 = 3^1 -> We need the weight of the next power of 3 for the weight of 3 kg.
9 = 3^2 -> We need the weight of the next power of 3 for the weight of 9 kg.
27 = 3^3 -> We need the weight of the next power of 3 for the weight of 27 kg.
Now, we have understood the pattern. We can use 1 kg, 3 kg, 9 kg, and 27 kg weights to weigh any load from 1 to 40 kg.
For example, to weigh 2 kg, we can put the 1 kg weight on one side of the scale and nothing on the other side. It will balance, indicating that the load weighs 2 kg. Similarly, we can use combinations of these weights to weigh any load within the 1-40 kg range.
Therefore, the weights should weigh 1 kg, 3 kg, 9 kg, and 27 kg each, respectively.