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?

Work rates per hour: X = 1/6, Y = 1/8, Z = 1/8. Since only one can work in any given hour and nobody can work two consecutive hours, we alternate so that the fastest worker (X) does as many hours as possible. In a sequence like X, Y, X, Z, X, Y, X (7 slots), X works in 4 non-consecutive hours and Y,Z fill the gaps. After 6 hours pattern X-Y-X-Z-X-Y, work done = 3·(1/6) + 1·(1/8) + 1·(1/6) ... computing optimally yields the work completed after 6 hours leaving a small fraction that requires a 7th slot but only part of it. Working out the optimal schedule X(1/6)+Y(1/8)+X(1/6)+Z(1/8)+X(1/6)+Y(1/8)+X(remaining) gives total time 6 hours 45 minutes for finishing the remaining fraction within X's next slot.