Rachel: - takes 1 minute to cross
Ben: - takes 2 minutes to cross
George: - takes 7 minutes to cross
Yvonne: - takes 10 minutes to cross
You can accomplish strategy 2 by having Yvonne and Rachel cross to the other side, taking 10 minutes. Rachel goes back and gets George, altogether taking 8 minutes. Again, Rachel goes back to get Ben, making 3 minutes. Altogether that journey would take 21 minutes.
However, the optimum method is when Rachel and Ben travel first and Rachel returns back to where she had started with the torch. Then, George and Yvonne go across. Ben is already on the other side but he should cross the bridge to get to Rachel. Finally, Rachel and Ben should cross the bridge together. The sum follows 2 minutes + 1 minute + 10 minutes+ 2 minutes + 2 minutes= 17 minutes.
Strategy 1 (17 minutes) involves taking the top two quickest people across, have the quickest come back, take the two slowest people across, have the quickest person on the destination side of the bridge return with the torch to bring the quickest person overall.
Strategy 2 (21 minutes) involves taking the person who takes the longest with the person who takes the shortest amount of time. The quickest person would then return to take the second-slowest person across the bridge, come back and then take the third-slowest (second-quickest) person across the bridge.
Strategy 1 is the most efficient when the time each person takes is more random, and often with a higher range (meaning it is less consistent).
Strategy 2 is more efficient when the timings are more consistent (likely with a lower range).
For example, with the times 7 minutes,12 minutes,17 minutes and 24 minutes, strategy 1 works to produce the quickest total time.
Meanwhile, with the times 1 minute, 2 minutes, 3 minutes and 4 minutes, strategy 2 works best.
The times 8 minutes, 14 minutes, 20 minutes and 26 minutes also work to prove this, with consistent 6 minute differences between each person’s time, as strategy 2 achieves the quickest total time here.
When all 4 people take the same amount of time, or just one person takes a different amount of time, both strategies are equally as efficient, and take the shortest amount of time.