Twelve soldiers had to get to a place twenty miles distant as quickly as possible and each had to arrive at the same time. They requisitioned the services of a man with a small car.

“Through this zone I can do a constant 20 miles an hour, but I cannot carry more than four men at a time. At what rate can you walk?”
“All of us can do a steady four miles an hour.”
“Very well, then I will go ahead with four men, drop them somewhere on the road to walk, then will pick up four more (who will already have started out), drop them also, and return for the last four. So all you have to do is keep walking while you are on your feet and I will do the rest.”

They started at noon and arrived safely, together, at the same time, according to plan.

What was the exact time they arrived?

In order for all the soldiers to arrive at the destination at the same time, each must spend the same fraction of that time (and therefore number of hours) in the car as opposed to walking.

Let x be the distance the car takes the first (and in fact each) group of men.  This takes x/20 hours, during which the other soldiers travel x/5 miles, leaving a 4x/5 mile gap between them and the car. That gap is closed at the rate of 24 mph, and so takes (4x/5)/24 = x/30 hours. At 4 mph the soldiers have covered another x/7.5 miles and so are x/5+x/7.5 miles from the origin at this time. The car then takes them another x miles, leaving them at 4x/3 miles from the origin.

When the car has discharged the second group, the last group will have advanced another x/5 miles, meaning they're at x/5+x/7.5+x/5 miles from their origin, or 8x/15 miles, while the car is at x/5+x/7.5+x = 4x/3 miles from the origin.

The gap between the car and the last group is therefore 4x/3 - 8x/15 = 4x/5. Of this gap, the marching soldiers make up 1/6, or 2x/15 miles.  Therefore the last group of soldiers has marched 8x/15 + 2x/15 = 2x/3 miles.

But this has to be the same distance as travelled on foot by the first group, which was 20 - x miles.  So 2x/3 = 20 - x, or 5x/3 = 20, so x = 12. Each group is driven 12 miles and walks 8 miles, taking 12/20 + 8/4 = 2.6 hours.

Posted by Charlie on 2006-03-31 14:20:19