Comment by gus_massa

OK, I give up for now for 12. Let's try 4. I hope I can do 4 at least.

The clock has only the numbers 0, 3, 6, 9 (I replaced 12 with 0 to save a character.)

[spoiler alert] Initially the state is [0]--- After the first move it goes to 3 or 9.

Case 3) Let's assume it first move to 3, so the state is now 0[3]--

There is a 50% chance that it goes to 6 and we get 03[6]- so the answer is 9

There is a 50% chance that it goes to 0 again, and we have [0]3--.

- Now there is a 50% chance that it goes to 9 and we get 03-[9] and the answer is 6.

- Now there is a 50% chance that it goes to 3 back and we get 0[3]--.

So starting from 0[3]--, there is a 50% of "9", 25% of "6", and 25% of back to 0[3]--. Using standard series trick, we can split the 25% loop in the final answers "9" and "6", in the same proportion of 50% vs 25%.

So the net result is that starting from 0[3]--, there is a 66.6...% of "9", 33.3...% of "6".

Case 9) Let's assume it first move to 9, so the state is now 0--[9] yada yada there is a 66.6...% of "3", 33.3...% of "6".

---

In total: (66.6%+0%)*50% = 33.3...% for 3 and the same for 9, and (33.3..%+33.3...%)*50%=33% for 6. So all endings are equiprobable. I still can believe my eyes, and I'm afraid there is an "obvious" solution that I'm missing.