X, Y and Z can complete a piece of work individually in 6 hours, 8 hours and 8 hours respectively. However, only one person at a time can work in each hour and nobody can work for two consecutive hours. All are engaged to finish the work. What is the minimum amount of time that they will take to finish the work?

The correct option is 6 hours 45 minutes.

To find the minimum time required to complete the work, we need to maximize the amount of work done in each hour under the given constraints.

Let us assume the total work is the Least Common Multiple (LCM) of the individual times taken by X, Y, and Z. The LCM of 6, 8, and 8 is 24. Therefore, let the total work be 24 units.

Now, we can calculate the individual work rates (efficiency) per hour for X, Y, and Z:
$$\text{Rate of X}=\frac{24\text{units}}{6\text{hours}}$$
$$\text{Rate of Y}=\frac{24\text{units}}{8\text{hours}}$$
$$\text{Rate of Z}=\frac{24\text{units}}{8\text{hours}}$$

To finish the work in the minimum amount of time, we must employ the most efficient worker, X, as much as possible. However, the constraints specify that:
1. Only one person can work in each hour.
2. No person can work for two consecutive hours.

To satisfy these constraints while maximizing efficiency, X should work in every alternate hour (e.g., 1st, 3rd, 5th, and 7th hours), while Y and Z alternate in the remaining hours (e.g., 2nd, 4th, and 6th hours).

Let us schedule the work hour-by-hour to calculate the cumulative work done:
• Hour 1: X works and completes 4 units.
• Hour 2: Y works and completes 3 units.
• Hour 3: X works and completes 4 units.
• Hour 4: Z works and completes 3 units.
• Hour 5: X works and completes 4 units.
• Hour 6: Y works and completes 3 units.

After 6 hours, the total work completed is:
$$4+3+4+3+4+3=21\text{units}$$

The remaining work is:
$$24-21=3\text{units}$$

In the 7th hour, X can work again since X did not work in the 6th hour. X completes 4 units of work per hour. Since only 3 units of work are left, the time taken by X to finish the remaining work is:
$$\text{Time}=\frac{3\text{units}}{4\text{units/hour}}$$

Converting $\frac{3}{4}$ of an hour into minutes:
$$\frac{3}{4}\times 60\text{minutes}=45\text{minutes}$$

Thus, the total minimum time required to finish the work is 6 hours and 45 minutes.