1 set contains: 3, 2, 1
2nd set: 6, 6, 7
3rd set: 10, 9, ?
What is ?
Choose from: 7, 8, 11, 12, 13
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Perhaps it is
if you take the sum of each circle you get 6, 19, and 19 + ?. If you add them together in pairs, you get 25, 38 + ?, and 25 + ?. 25 is a perfect square. If ? = 11, then you get 25, 36, and 49 for the sums, all perfect squares (and 5, 6, 7 which is in order and slightly comforting).
One thought I had, although I have no idea how it would lead to the solution, is that:
All three large circles could actually be the same circle, which is concealed by a rotating disc with three holes. So it would actually be like a clock face, with the numbers 9 2 6 _ 10 1 7 _ ? 3 6 _ appearing in clockwise order, starting from the top.
Another vote for
I don't think we have enough numbers/circles to establish a satisfactory pattern, but this is what I went for.
Look at the top left 3 numbers. It's like an arrow with the 3 pointing west. For the circle to its right (clockwise) it's like that arrow has rotated clockwise juuuust a little bit. Then the 3 is replaced by 6, the 2 by the other 6 and 1 by the 7. For the bottom circle you rotate that arrow a little bit again and you replace one 6 with a 9 and the other with 10. Therefore, the patterns are:
3 -> 6 -> 9 (+3)
2 -> 6 -> 10 (+4)
1 -> 7 -> ? (+6)
After posting I realised that
my answer is effectively the same as Gordon's, because you increase the sum of each circle by 13 (3 + 4 + 6). So, we can't pool our answers to say 13 is more likely than other guesses.