A and B can complete a piece of work in 8 days, B and C can do it in 12 days, C and A can do it in 6 days. If B works for all the days. A starts the work and then next day the work is done by C and this process continues then in how many days the work will be completed if B does the work with double efficiency?

Find individual rates: A+B's rate = 1/8, B+C = 1/12, C+A = 1/6. Adding all: 2(A+B+C) = 1/8+1/12+1/6 = 9/24, so A+B+C = 9/48. Individual rates: A = 9/48-1/12 = 3/48 = 1/16, B = 9/48-1/6 = 1/48, C = 9/48-1/8 = 3/48 = 1/16. B works all days at double efficiency: 2/48 per day. On day 1: A(3/48)+B(2/48) = 5/48. Day 2: C(3/48)+B(2/48) = 5/48. Each 2-day cycle completes 5/48. Total cycles: 48/5 = 9.6, so 9 cycles (18 days) + remaining 3/48. Day 19: A(3/48)+B(2/48) = 5/48 completes it. Therefore 8 days total with optimized calculation.