The camel and the bananas
You have 3000 bananas and a camel, and the bananas must be taken to a market 1000 km away across a desert. The camel carries at most 1000 bananas at a time and eats one banana for every kilometre it walks, in either direction (it must carry what it eats). You may leave bananas in depots along the way and pick them up later.
Treat distances and banana counts as continuous. What is the largest number of bananas that can reach the market?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
To move 3000 bananas forward one kilometre, the camel must cross that kilometre several times. How many?
The cost per kilometre drops each time the stock falls to the next multiple of 1000. Move the stock forward in stages.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Moving a stock of bananas forward costs one banana per kilometre for every crossing of that kilometre. With more than 2000 bananas the camel needs 3 loads, so 5 crossings per kilometre; with more than 1000, 3 crossings; with at most 1000, one crossing. Shift the whole stock forward until it drops to the next multiple of 1000, then continue with fewer trips.
Stage 1: from 3000 to 2000. Three loads mean three trips forward and two back: 5 bananas per kilometre. The stock reaches 2000 after
Stage 2: from 2000 to 1000. Two loads: two trips forward and one back, 3 bananas per kilometre. The stock reaches 1000 after another at km from the start.
Stage 3: the last stretch. One load of 1000 bananas, one trip of km, eating one banana per kilometre. What arrives is
Why this is best. As long as more than 2000 bananas remain, three loads are unavoidable, so each kilometre costs at least 5; the plan never pays more than the minimum number of crossings at any stage, and it reduces the number of loads as early as possible.
With whole kilometres the same plan delivers 533 bananas.