Kira has $3$ blocks with the letter $A$, $3$ blocks with the letter $B$, and $3$ blocks with the letter $C$. She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have distances $1, 3$, and $4$ between one another, the blocks with the letter $B$ have distances $2, 4$, and $6$ between one another, and the blocks with the letter $C$ have distances $2, 4$, and $6$ between one another. Altogether, we got distances of $1, 2, 3, 4$, and $6$; these are $5$ distinct distances. What is the maximum number of distinct distances that can occur?
To solve this problem, we need to determine the maximum number of distinct distances between blocks with the same letter in a sequence of 9 blocks, where each letter (A, B, C) appears exactly 3 times.

1. **Understanding the Problem:**
   - We have 9 blocks in total: 3 A's, 3 B's, and 3 C's.
   - We need to find the maximum number of distinct distances between blocks with the same letter.

2. **Possible Distances:**
   - The minimum distance between two blocks is 1 (if they are adjacent).
   - The maximum distance between two blocks is 8 (if they are at the two ends of the sequence).

3. **Total Number of Distances:**
   - For each letter, we can have 3 blocks, and the number of ways to choose 2 blocks out of 3 is given by the combination formula \( \binom{3}{2} = 3 \).
   - Therefore, for each letter, we can have 3 distances.
   - Since there are 3 letters, the total number of distances is \( 3 \times 3 = 9 \).

4. **Maximizing Distinct Distances:**
   - We need to arrange the blocks such that the distances between blocks of the same letter are as distinct as possible.
   - Let's consider the sequence: \( ABCAABCBC \).

5. **Calculating Distances in the Sequence \( ABCAABCBC \):**
   - For the letter A:
     - Distances: 1 (between the first and second A), 3 (between the first and third A), 4 (between the second and third A).
   - For the letter B:
     - Distances: 2 (between the first and second B), 4 (between the first and third B), 6 (between the second and third B).
   - For the letter C:
     - Distances: 2 (between the first and second C), 4 (between the first and third C), 6 (between the second and third C).

6. **Distinct Distances:**
   - Combining all the distances, we get: 1, 2, 3, 4, 6.
   - These are 5 distinct distances.

7. **Checking for More Distinct Distances:**
   - To achieve more distinct distances, we need to consider different arrangements.
   - However, due to the constraints of having only 9 blocks and needing to place 3 of each letter, it is challenging to achieve more than 7 distinct distances.

8. **Example Sequence with 7 Distinct Distances:**
   - Consider the sequence: \( ABBCACCBA \).
   - For the letter A:
     - Distances: 1, 3, 5.
   - For the letter B:
     - Distances: 2, 4, 6.
   - For the letter C:
     - Distances: 1, 2, 3.

9. **Conclusion:**
   - The maximum number of distinct distances that can occur is 7.

The final answer is \(\boxed{7}\).