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?

• X, Y, and Z complete 1/6, 1/8, and 1/8 of the work per hour respectively, with X being the fastest worker.

• To minimize time, the fastest worker (X) should work as many hours as possible, subject to the constraint that no one works two consecutive hours.

• In a 6-hour cycle, the optimal pattern is X, Y, X, Z, X, Y (or similar), where X works 3 hours, Y works 2 hours, and Z works 1 hour.

• Work done in 6 hours: 3(1/6) + 2(1/8) + 1(1/8) = 1/2 + 3/8 = 7/8 of the total work.

• Remaining work is 1/8. Since X just worked in the 6th hour (in the pattern X,Y,X,Z,X,Y), Y or Z must work next. Y works 1/8 in 1 hour, but we need to check if a faster combination exists or if the pattern allows X to work in the 7th hour. Wait, if the pattern is X, Y, X, Z, X, Y, the 6th hour is Y. So X can work in the 7th hour.

• Let's re-evaluate the optimal sequence for minimum time. We want to maximize X's hours. Sequence: X, Y, X, Z, X, Y. Hours: 1(X), 2(Y), 3(X), 4(Z), 5(X), 6(Y). Work: 3/6 + 2/8 + 1/8 = 1/2 + 3/8 = 7/8. Remaining: 1/8.

• Next hour (7th): X can work (since Y worked in 6th). X does 1/6 in 1 hour. But we only need 1/8. Time taken = (1/8) / (1/6) = 6/8 = 0.75 hours = 45 minutes.

• Total time = 6 hours + 45 minutes = 6 hours 45 minutes.