A trader has 3000 bananas and a camel that carries at most 1000 at a time and eats 1 banana per kilometre travelled in either direction. The market is 1000 km away and bananas may be left in piles along the road. What is the largest number that can be delivered?

A trader has 3000 bananas and a camel that carries at most 1000 at a time and eats 1 banana per kilometre travelled in either direction. The market is 1000 km away and bananas may be left in piles along the road. What is the largest number that can be delivered?

Approach: Work out how many crossings of each kilometre are needed as a function of how many bananas remain, and note that the rate drops each time the stock falls below a multiple of the camel's capacity.

533. Above 2000 bananas the camel needs 5 crossings of each kilometre, three forward and two back, so the stock falls by 5 per km. Dropping from 3000 to 2000 therefore takes 200 km, leaving 2000 bananas at the 200 km mark. Between 2000 and 1000 only 3 crossings per kilometre are needed, so 1000 bananas cover 1000/3 = 333.33 km, leaving 1000 bananas at 533.33 km. From there one load of 1000 makes a single trip over the remaining 466.67 km and arrives with 533.33, so 533 whole bananas. Carrying the stock in three separate straight runs delivers nothing, since a one way trip of 1000 km eats a full load of 1000.

Follow-up: How does the answer change if the market is 800 km away instead of 1000 km?

Key concepts: crossing count, stock depletion rate, one way trip.