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In order to reach their goal, it is necessary for the explorers to delay eating until the end of each day. This is reasonable if one assumes that the explorers would break camp after a hearty breakfast. We also assume that upon departure the explorers emptied their base camp of all provisions, so that upon returning they will only be able whatever they have carried with them before being evacuated.
At the end of each day's march the explorers consume their daily ration, and half of the remaining provisions are then cached on site. Two thirds of the explorers present then head back, empty-handed, in the direction of the base camp, consuming provisions stored on previous days along their way. The remaining explorers replenish their supplies so they are again carrying a full three days provisions and continue their journey. In this way, the team of explorers can send one of their number the required distance, with all returning safety to the base camp.
An alternative interpretation of the problem would have the explorers eating continuously from the moment they break camp. Here, a team of K explorers can reach a maximum distance 3K/{K + 1} from the base camp. To achieve this, a single explorer will have to turn back at each of the points 3J/{K+1} for J=1,..K; dropping off whatever food he is carrying in excess of his needs and replenishing his provisions as passes each turn back point on the way home. Notice that in this interpretation the explorers can never get beyond three days march from camp, no matter how large their expedition.
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