Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
1. Let's denote the work rates of Barbara, Edward, Abhinav, and Alex as follows:
   - Barbara: $\frac{1}{10}$ of the test per day
   - Edward: $\frac{1}{9}$ of the test per day
   - Abhinav: $\frac{1}{11}$ of the test per day
   - Alex: $\frac{1}{12}$ of the test per day

2. Since only one person works on the test per day and they take turns, we consider cycles of 4 days. In each cycle, the total amount of work done is the sum of their individual work rates:
   \[
   \frac{1}{10} + \frac{1}{9} + \frac{1}{11} + \frac{1}{12}
   \]

3. To find the sum of these fractions, we need a common denominator. The least common multiple (LCM) of 10, 9, 11, and 12 is 11880. Converting each fraction to have this common denominator, we get:
   \[
   \frac{1}{10} = \frac{1188}{11880}, \quad \frac{1}{9} = \frac{1320}{11880}, \quad \frac{1}{11} = \frac{1080}{11880}, \quad \frac{1}{12} = \frac{990}{11880}
   \]

4. Adding these fractions together:
   \[
   \frac{1188}{11880} + \frac{1320}{11880} + \frac{1080}{11880} + \frac{990}{11880} = \frac{4578}{11880}
   \]

5. Simplifying the fraction:
   \[
   \frac{4578}{11880} = \frac{381}{990} = \frac{127}{330}
   \]

6. Therefore, in one cycle of 4 days, $\frac{127}{330}$ of the test is completed. To find out how many cycles are needed to complete the test, we solve:
   \[
   n \cdot \frac{127}{330} \geq 1
   \]
   where \( n \) is the number of cycles. Solving for \( n \):
   \[
   n \geq \frac{330}{127} \approx 2.598
   \]
   Thus, at least 3 cycles are needed.

7. In 3 cycles (12 days), the amount of work done is:
   \[
   3 \cdot \frac{127}{330} = \frac{381}{330} = 1.1545
   \]
   This means the test is completed in 12 days, with some extra work done.

The final answer is $\boxed{12}$ days.