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The problem asks for the minimum number \( n \) of stones required to measure all integer quantities from 1 kg to 121 kg, where the stones can be placed on either side of the balance.

This is a classical problem related to representing all integers within a range using powers of 3. The reason for powers of 3 is that placing stones on either side of the balance allows three states for each stone: the stone can be placed on the left pan, right pan, or not used at all. This corresponds to a base-3 system (ternary system) where each coefficient can be -1, 0, or 1.

### Explanation:

We need to express numbers from 1 kg to 121 kg using stones of weights in powers of 3, because of the three possible positions (left, right, or not used).

The powers of 3 are:

\[
3^0 = 1, \quad 3^1 = 3, \quad 3^2 = 9, \quad 3^3 = 27, \quad 3^4 = 81, \quad 3^5 = 243, \quad \dots
\]

We need the minimum number \( n \) such that the sum of these powers covers the range from 1 to 121 kg.

Start by calculating the cumulative sum of powers of 3:

\[
1 + 3 + 9 + 27 + 81 = 121
\]

Thus, using 5 stones with weights \( 1, 3, 9, 27, 81 \) is enough to represent all weights from 1 kg to 121 kg, using the property of placing stones on either side of the balance.

### Conclusion:
The minimum value of \( n \) is 5, so the correct answer is:

\[
\boxed{5 \text{ (C)}}
\]

Would you like more details on how the ternary system works for this problem or anything else?

### Related Questions:
1. How do you determine the required powers of 3 in such problems?
2. Can this problem be solved using any other base system, and why or why not?
3. What is the mathematical reasoning behind choosing powers of 3 for this problem?
4. How does placing stones on either side of the balance simplify the solution?
5. What would be the solution if the range was increased beyond 121 kg?

### Tip:
Whenever dealing with weighing problems, consider the power system (usually 2 or 3) for efficient representation across a range of values.

A retailer has n stones by which he can measure (weight) all the quantities from 1 kg to 121 kg (in integer only), keeping these stones on either side of the balance. What is the minimum value of n?

Learn how to solve the problem of determining the minimum number of stones needed to measure all quantities from 1 kg to 121 kg using the powers of 3. Understand key concepts in number theory and the ternary system. Suitable for Grades 10-12.

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