In the days before the invention of aeroplanes European explorers faced grave difficulties exploring the foodless and nearly impenetrable jungles of New Guinea.

With time and experience, they learnt that a well trained and physically fit explorer could carry a three days supply of food with him.

Show how a team of 3^N-1 explorers could send one of their number a distance N days march from the base camp, with all explorers returning safely to the base camp. (Note that once an explorer returns to camp, he must stay there.)

If the consumption of food is a quanta, calculated per day, as opposed to a fluidic consumption, the travel an explorer makes is limited...

For 3^1-1 = 1 explorer, the explorer can travel 2 days out.
• E₁ travels 1 day out (checkpoint 1), consuming 1 day of food.
• E₁ continues a 2nd day out and consumes a 2nd day of food.
• E₁ returns to checkpoint 1 and consumes a 3rd day of food.
• E₁ returns to base camp (where he may raid the kitchen).

For 3^2-1 = 3 explorers, one explorer can travel 3 days out.
• E₁, E₂ and E₃ travels 1 day out (checkpoint 1), consuming 1 day of food.
• E₁ transfers 1 day food to E₂ and E₃.
• E₁ returns to base camp.
• E₂ and E₃ travels a 2nd day out (checkpoint 2), consuming a 2nd day of food.
• E₂ transfers 1 day food to E₃.
• E₂ returns to checkpoint 1, consuming a 3rd day of food.
• E₃ returns to base camp.
• E₃ travels a 3rd day out, consumes a 3rd day of food.
• E₃ returns to checkpoint 2 and consumes a 4th day of food.
• E₃ returns to checkpoint 1 and consumes a 5th day of food.
• E₃ returns to base camp.

For a fluidic consumption, where food may be cached and transferred (at each "checkpoint", one explorer caches an amount needed to return to base camp for all other explorers and transfers the same amount to each explorer, returning each explorer to a full load, and keeping enough for his own return to base camp) the distance one explorer may reach with 3^N-1 explorers setting out and returning to base camp is as follows:
(for x=1 to 3^N-1) SUMMATION: 3/(2x)

• For 3^1-1 = 1 explorer, the explorer can travel 1 1/2 days out.
• For 3^2-1 = 3 explorers, one explorer can travel 2 3/4 days out.
• For 3^3-1 = 9 explorers, one explorer can travel approximately 4 1/4 days out.
• For 3^4-1 = 27 explorers, one explorer can travel approximately 5 5/6 days out.
• For 3^5-1 = 81 explorers, one explorer can travel approximately 7 1/2 days out.
• For 3^5-1 = 243 explorers, one explorer can travel approximately 9 1/9 days out.
• etc....

Posted by Dej Mar on 2009-05-08 05:16:17