So how the balanced ternary numeral system can help us to solve this problem?
Basically, as was mentioned in previous posts, it uses to represent numbers three digits {1,0,1} which have a "natural" interpretation of measurement when we weigh unknown cargo Q with "ternary" weights, but let's go through it point by point.
Suppose, we have a 4 ternary weights 1 kg, 3 kg, 9 kg, 27 kg (3^{0}, 3^{1}, 3^{2}, 3^{3}) and suppose that our cargo Q takes the value of the natural numbers, that is, {1, 2, 3, ...}. Then for four ternary weights maximum load is the sum of all ternary weights, i.e. 40 = 1 + 3 + 9 + 27.
Weighing begins with top weights, and the result of the weighing is represented as a 4places "ternary" number.
It is clear that for weighing load Q = 1 kg, it suffices to one weight of 1 kg which is placed on the right (that is, free from the load of Q) scales. Digital recording of this weighing is as follows:
1 kg
Ternary
representation of decimal number 1

0

0

0

1

weights

27kg

9 kg

3 kg

1 kg

Weighing load Q = 2 kg will require the use of two weights: first, to "empty" bowl placed a weight of 3 kg and then a bowl with a load  1 kg. The result of this weighing written in the form:
2 kg
Ternary
representation of decimal number 2

0

0

1

1

weights

27kg

9 kg

3 kg

1 kg

Note that in this measurement, ternary records appeared "negative" one 1, which reflects the fact that the corresponding weight is on the left side of the scale (i.e., on the bowl with load).
Clearly, the "ternary" entry for goods in 3 and 4 kg, respectively, have the following view: 3 kg = 0010 and 4 kg = 0011. It is more difficult express shipping 5 kg = 0111. Loads 6, 7, 8, 9, 10 kg have the following ternary records:
6 kg
Ternary
representation of decimal number 6

0

1

1

0

weights

27kg

9 kg

3 kg

1 kg

7kg
Ternary
representation of decimal number 7

0

1

1

1

weights

27kg

9 kg

3 kg

1 kg

8kg
Ternary
representation of decimal number 8

0

1

0

1

weights

27kg

9 kg

3 kg

1 kg

9 kg
Ternary
representation of decimal number 9

0

1

0

0

weights

27kg

9 kg

3 kg

1 kg

10 kg
Ternary
representation of decimal number 10

0

1

0

1

weights

27kg

9 kg

3 kg

1 kg

and so on.
In general case, the npositions representation of an integer decimal number N in the balanced ternary numeral system is as follows:
where Ci is the one of ternary digits {1,0,1}.
For Example, number 7 can be represented in ternary as: 0×3^{3} + 1×3^{2}  1×3^{1} + 1×3^{0} .
This is just one of the remarkable properties and practical applications of balanced ternary system, about others we well talk in the next posts.