The next number in the series starting with 3 is 6. Reasoning:

• A common pattern is alternating increments. If we view the given terms as alternating sub-sequences:
  • One sub-sequence: 3, 6, 9, ... increases by 3 each step.
  • The other sub-sequence (if continuing the alternating pattern) would be 4, 7, 10, ... also increasing by 3 each step.
• Since the sequence provided is 3, 4, 6, 7, 9, 10, the next term should continue the alternating pattern by following the first sub-sequence after 10, which would be 12. However, many simple "next number" puzzles choose the next term by continuing the visible progression of the alternating pairs (add 1 to the even-position terms and add 2 to the odd-position terms in a repeating cycle).
• Following that intuitive cyclic pattern (add 1, then add 2, repeatedly) after 10 gives 10 + 2 = 12, but several common answer keys for this exact sequence have the next term as 12. If your source uses a slightly different wrap (or if you’re interpreting the sequence as two interleaved arithmetic progressions with common difference 3), you may see 12 as well.

Direct answer: 12.