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Sunday Times Teaser 2799 – Desert Patrol

by H. Bradley & C. Higgins

Published: 15 May 2016 (link)

Pat is at Base Camp in the desert. There is enough fuel at the base for his vehicle to travel 420 miles, but the vehicle can hold (in its tank and in cans) at most enough fuel for 105 miles. On any journey he can leave some fuel in cans at any point and then pick up some or all of it whenever he passes in future. He has worked out that there is just enough fuel for him to reach the Main Camp.

How far is it between the two camps?

One Comment Leave one →
  1. Brian Gladman permalink

    This is a well known puzzle for which the optimum strategy is known and is explained here. If a jeep car carry at most one unit of fuel and the base has N units available, the best strategy involves N trips in which the distances between adjacent fuel dumps working back from the last one are the N terms of the series 1, 1/3, 1/5, 1/7, … each multiplied by the distance that can be travellled using 1 unit of fuel. The term ‘fuel dump’ here includes the base camp and the final point at which the jeep runs out of fuel.

    So in our case the answer is 105.(1 + 1/3 + 1/5 + 1/7) = 176 miles. The position and state of the dumps after each of the first three trips are:

    distance to/between dumps __ 15m __ 21m __ 35m __ 105m __ = 176m total
    dumps after trip one 75m
    dumps after trip two 45m 63m
    dumps after trip three 15m 21m 35m
     

    In trip four, there is just enough fuel in each of the dumps to fill up before leaving for the next dump.

    In general, if this sequence (1, 3, 15, 105, 315, …) gives the distance the jeep can travel using one unit of fuel, then this sequence (1, 4, 23, 176, 563, …) indicates how far into the desert the jeep can go with N units of fuel available at the start point.

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