# Measuring Kilograms Puzzle - Solution

### The Puzzle:

In front of you is a balance.

Can you devise a system, using only 4 weights, that can measure any whole number of kilograms (1,2,3, etc) ...

a) ... up to 15 kg ?
b) ... up to 40 kg ?
c) ... up to 80 kg ?

### Our Solution:

a) Up to 15 kg:

Weights are: 1 kg, 2 kg, 4 kg, 8 kg

Like this:

1 kg = 1 kg
2 kg = 2 kg
3 kg = 2 kg + 1 kg
...
14 kg = 8 kg + 4kg + 2 kg
15 kg = 8 kg + 4 kg + 2 kg + 1 kg

b) Up to 40 kg:

Weights are: 1 kg, 3 kg, 9 kg, 27 kg

Like this:

You can put the weights on the other side of the balance, therefore subtracting some of the weight from the other side. Using this system the objects can be weighed thus:

1kg = 1kg
2kg = 3kg - 1kg
3kg = 3kg
...
39kg = 27kg + 9kg + 3kg
40kg = 27kg + 9kg + 3kg + 1kg

c) Up to 80 kg:

Weights are: 2 kg, 6 kg, 18 kg, 54 kg

Like this:

Put 2kg on one side, your item on the other. If item is less then 2kg, it is 1 kg.
If it is more, put 6kg on one side, your item and 2kg on the other. If item and 2kg are less then 6kg, item is 3kg.
If it is more, put 6kg on one side, your item on the other. If item is less then 6kg, it's 5kg.
...
etc, up to 80 kg

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