NL-NWO 2022 Problem 5

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?