Four friends A, B, C and D need to cross a bridge. A maximum of two persons can cross it at a time. It is night and they just have one lamp. Persons that cross the bridge must carry the lamp to find the way. A pair must walk together at the speed of slower person. After crossing the bridge, the person having faster speed in the pair will return with the lamp each time to accompany another person in the group. Finally, the lamp has to be returned at the original place and the person who returns the lamp has to cross the bridge again without lamp. To cross the bridge, the time taken by them is as follows : A: 1 minute, B: 2 minutes, C: 7 minutes and D: 10 minutes. What is the total minimum time required by all the friends to cross the bridge ?

Correct Option: (a) 23 minutes

To minimize crossing time while adhering to the rule that the faster person in the pair returns with the lamp: 1. A and B cross the bridge. Time taken = 2 minutes (speed of slower). A (1 min) being faster than B (2 min) returns with the lamp. Time = 1 minute. (Cumulative time = 3 minutes). 2. A and C cross. Time taken = 7 minutes. A (faster) returns with the lamp. Time = 1 minute. (Cumulative time = 11 minutes). 3. A and D cross. Time taken = 10 minutes. (Cumulative time = 21 minutes). All four are now across, but the lamp is on the far side. 4. The lamp must be returned to the original side. A (the fastest) takes the lamp back. Time taken = 1 minute. (Cumulative time = 22 minutes). 5. Finally, A has to cross back without the lamp. Time taken = 1 minute. Total minimum time = 22 + 1 = 23 minutes. Other options like 20, 21, or 22 minutes are mathematically insufficient to satisfy all constraints.