A set of data has observations such that no two observations are equal. The median of this data is 12. A number is added to the data and the median remains 12. Find the number.

Answer

To solve the problem step by step, we will follow the logical reasoning based on the information given about the data set and the properties of the median.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have a data set with observations such that no two observations are equal.
- The median of the original data set is 12.
- A number is added to this data set, and the median remains 12.

2. **Determine the Number of Observations**:
- Since the median is given, we can infer that the total number of observations is even. Let's assume there are 12 observations (as an example, since the actual number isn't specified but is implied to be even).

3. **Identify the Median in an Even Set**:
- For an even number of observations, the median is calculated as the average of the two middle numbers.
- If we denote the observations as \(x_1, x_2, x_3, \ldots, x_{12}\), the median is given by:
\[ \text{Median} = \frac{x_6 + x_7}{2} = 12 \]
- This implies:
\[ x_6 + x_7 = 24 \]

4. **Add a New Number to the Data Set**:
- When we add a new number \(N\) to the data set, the total number of observations becomes 13 (which is odd).
- The median of an odd number of observations is the middle observation, which is the 7th observation in this case.

5. **Determine the New Median**:
- The new median remains 12, which means the 7th observation must also be 12:
\[ x_7 = 12 \]

6. **Relate the Observations**:
- From the previous step, we have \(x_6 + x_7 = 24\) and \(x_7 = 12\).
- Therefore, substituting \(x_7\) into the equation gives:
\[ x_6 + 12 = 24 \implies x_6 = 12 \]
- However, since no two observations can be equal, we need to adjust our observations. Thus, we can assume:
- Let \(x_6 = 11\) and \(x_7 = 13\) (which are valid since they are not equal).

7. **Conclusion**:
- The new number \(N\) that can be added to the data set must be 12, as it fits between \(x_6\) and \(x_7\) (11 and 13), ensuring that the median remains 12.

### Final Answer:
The number that is added to the data set is **12**.