Enigma 550: Do your level best
From New Scientist #1703, 10th February 1990 [link]
Bern, Cara and Loren are climbing in the Hillymayas and have decided to tackle Mount Veryrest which is 700 metres high. There are three different approaches and they each take a different one. Bern’s approach is shown in the diagram.
All slopes in the Hillymayas have the same steepness — 1 up or down for 1 across; all climbers go at the same pace — 100 metres up or down in each hour. Thus, Bern could reach the top in 13 hours.
Bern’s approach is described as: up 5, down 3, up 5.
Cara’s approach is: up 3, down 2, up 4, down 1, up 3.
Loren’s approach is: up 5, down 2, up 3, down 6, up 7.
The three decide that, to be fair, they will climb so that, at each point in time, all three are at the same height. This will involve some retracing of steps. Given that condition, they try to reach the top as soon as they can.
How may hours does it take them to reach the top?