A is half as efficient as B who is thrice as efficient as C. In how many days can all $3$ of them complete a task if B can complete the task alone in $6$ days?

Let the efficiency of C be $x$.
Then, the efficiency of B = $3x$.
The efficiency of A = $\frac{1}{2}\times 3x=1.5x$.

The ratio of efficiencies of $A:B:C=1.5x:3x:x=3:6:2$.

Given that B alone can complete the task in $6$ days:

$$\text{Total Work}=\text{Efficiency of B}\times \text{Days}=6\times 6=36\text{ units}$$

The combined efficiency of A, B, and C:

$$\text{Combined Efficiency}=3+6+2=11\text{ units/day}$$

Number of days required by all three to complete the task:

$$\text{Days}=\frac{36}{11}\approx 3.27\text{ days}$$