8105 has one bull and one cow. I believe that narrows it down to only one possibility, though it’s entirely possible that I’m mistaken.
| # | guess | bulls | cows |
|---|---|---|---|
| 1 | 1234 | 0 | 1 |
| 2 | 5678 | 0 | 2 |
| 3 | 0912 | 0 | 2 |
| 4 | 9056 | 0 | 2 |
| 5 | 7195 | 2 | 0 |
| 6 | 7520 | 0 | 2 |
| 7 | 8105 | 1 | 1 |
3476 has no bulls and no cows.
| # | guess | bulls | cows |
|---|---|---|---|
| 1 | 1234 | 0 | 1 |
| 2 | 5678 | 0 | 2 |
| 3 | 0912 | 0 | 2 |
| 4 | 9056 | 0 | 2 |
| 5 | 7195 | 2 | 0 |
| 6 | 7520 | 0 | 2 |
| 7 | 8105 | 1 | 1 |
| 8 | 3476 | 0 | 0 |
Same result: 3170 has no bulls and no cows.
| # | guess | bulls | cows |
|---|---|---|---|
| 1 | 1234 | 0 | 1 |
| 2 | 5678 | 0 | 2 |
| 3 | 0912 | 0 | 2 |
| 4 | 9056 | 0 | 2 |
| 5 | 7195 | 2 | 0 |
| 6 | 7520 | 0 | 2 |
| 7 | 8105 | 1 | 1 |
| 8 | 3476 | 0 | 0 |
| 9 | 3170 | 0 | 0 |
I wonder what the secret number is. 
Can anybody help us here?
We are still waiting for someone to guess the secret number.
In the meantime I have a different kind of secret number:
Here’s some information about it:
+ = +
4 - 7 = -
What is the number symbolized by the colored circles?
Or you can guess bibliophage’s secret number. 
0 0 0 0. What do I win?
Yeah, I forgot to mention that the same rules apply.
You can only use a digit once in this number.
You were close though.
The solution is not unique. I have found three solutions so far.
You’re too good. This was just a distraction. A way to bump this thread.
I don’t think there are enough people to sustain this idea, so I’ll just let it die now.
I guess I’m restless. I’ve come up with additional clues so that the solution should not be ambiguous anymore:
So, this is a a different kind of secret number:
Each digit must occur in this number only once.
Here is the complete (I hope) list of clues about it:
+ = +
4 - 7 = -
2 - 2 = 2
9 - 5 =
What is the number symbolized by the colored circles?
Or you can guess bibliophage’s secret number. 
8169?
8169? Please look at the first hint. Does 8+1 = 6+9 ? No, it doesn’t.