3 2 2 5 5 5 7 8 8 9 11 11 ?

Concept: A number sequence that looks irregular when read straight through is sometimes actually several separate arithmetic series interleaved together, one term from each series in rotation. To solve it, split the sequence into every nth term (here every 3rd term) and treat each split as its own independent arithmetic series with its own common difference.

1. Write out the sequence with position numbers: 3(1), 2(2), 2(3), 5(4), 5(5), 5(6), 7(7), 8(8), 8(9), 9(10), 11(11), 11(12), ?(13).

2. Group every 3rd term starting at position 1: positions 1, 4, 7, 10, 13 give 3, 5, 7, 9, ? — each term is 2 more than the one before (3, 5, 7, 9), so this series has a common difference of +2.

3. Group every 3rd term starting at position 2: positions 2, 5, 8, 11 give 2, 5, 8, 11 — a common difference of +3.

4. Group every 3rd term starting at position 3: positions 3, 6, 9, 12 give 2, 5, 8, 11 — also a common difference of +3.

5. Position 13 belongs to the first group (positions 1, 4, 7, 10, 13), so continue that series with its own step: 9 + 2 = 11.

Cross-check: the other two groups both settle into a clean +3 progression once separated out, confirming the three-series structure is real rather than coincidental. Applying the first group's own step (not the step belonging to the other two groups) to its last known term gives the missing value.