Four persons can cross a bridge in 3 minutes, 7 minutes, 13 minutes and 17 minutes respectively. At most two people can cross the bridge at a time, and whenever two cross together they must walk at the slower person's speed. A torch is required to cross, and someone must bring it back each time until everyone has crossed. What is the minimum total time required for all four persons to cross the bridge?

Introduction / Context:
This is a classic bridge-and-torch puzzle, commonly asked in aptitude tests. Each person takes a different time to cross, and only two can cross together. The key challenge is to schedule the crossings so that the total time is minimised while respecting the constraints on speed and the need to bring the torch back.

Given Data / Assumptions:

• There are four people with individual crossing times: 3 minutes, 7 minutes, 13 minutes, and 17 minutes.

• At most two people can be on the bridge at a time.

• When two people cross together, the crossing time equals the slower person's time.

• A torch is required on the bridge, so when two cross forward, one person must sometimes return with the torch.

• We assume the goal is to get all four to the other side in the minimum possible total time.

Concept / Approach:
We label the four people A (3 minutes), B (7 minutes), C (13 minutes) and D (17 minutes). Two common strategies exist: always send the two slowest together, or repeatedly send the fastest person back with the torch. We must compute and compare the total times of these strategies to find the minimum.

Step-by-Step Solution:
Step 1: Let A = 3, B = 7, C = 13, D = 17 (minutes). Step 2: Use the optimal standard strategy: 1) A and B cross first (time = 7 minutes). Now A and B are on the far side. Step 3: A returns with the torch (time = 3 minutes). Total time so far = 10 minutes. Step 4: C and D (the two slowest) cross together (time = 17 minutes). Total time so far = 27 minutes. Step 5: B returns with the torch (time = 7 minutes). Total time so far = 34 minutes. Step 6: Finally, A and B are already across, and C and D have crossed; everyone is now on the far side, and the total time is 34 minutes.

Verification / Alternative check:
Consider the alternative naive strategy: always pair the fastest (A) with one of the others: A + D cross (17), A returns (3), A + C cross (13), A returns (3), A + B cross (7). Total = 17 + 3 + 13 + 3 + 7 = 43 minutes. This is clearly worse than 34 minutes. Any other arrangement will end up being equivalent or worse than one of these strategies, so 34 minutes is indeed minimal.

Why Other Options Are Wrong:
Option A (17 minutes) is impossible because even the slowest pair alone takes 17 minutes, and we still need extra trips for others to cross and the torch to return.
Option B (20 minutes) is too small; just C and D crossing plus at least one return already exceed this time.
Option D (12 minutes) is unrealistic, as even sending the two slowest requires 17 minutes, so 12 minutes cannot be the total for four people.

Common Pitfalls:
Many candidates try to send the slowest person back with the torch or forget to count the return trips. Others do not systematically compare strategies and assume that sending the fastest person back every time is always best. The safe method is to write out each possible plan, compute the total time, and then choose the minimum.

Final Answer:
Thus, the least possible time required for all four persons to cross the bridge is 34 minutes.