Solution to: Camel & Bananas

The solution: 533⅓ bananas.

Explanation: Since there are 3,000 bananas and the camel can carry at most 1,000 bananas, at least five trips are needed to transport all the bananas from the plantation P (3 trips away from the plantation and 2 return trips):

Point A in the diagram cannot be the market. This is because the camel can never travel more than 500 kilometers into the desert if it is to return to the plantation (the camel eats a banana for every kilometer it travels!). Therefore, point A lies somewhere in the desert between the plantation and the market. From point A to the next point, fewer than five trips must be used to transport the bananas to that next point. We arrive at the following global solution to the problem (P denotes the plantation, M denotes the market):

Note that section PA must be included in the solution (as explained above), but section AB or section BM might have a length of 0. Let us now examine the costs of each part of the route. One kilometer on section PA costs 5 bananas. One kilometer on section AB costs 3 bananas. One kilometer on section BM costs 1 banana. To save bananas, we should ensure that the length of PA is less than the length of AB and that the length of AB is less than the length of BM. Since PA is greater than 0, we conclude that AB is greater than 0 and that BM is greater than 0.

The camel can carry away at most 2,000 bananas from point A. This means the distance between P and A must be chosen such that exactly 2,000 bananas arrive at point A. If PA is chosen to be smaller, more than 2,000 bananas would arrive at A, but the surplus cannot be transported further. If PA is chosen to be larger, we would lose more bananas to the camel than necessary. Now we can calculate the length of PA: 3000 - 5 × PA = 2000, so PA = 200 kilometers. Note that this distance is less than 500 kilometers, so the camel can travel back from A to P.

The situation at point B is similar to that at point A. The camel cannot transport more than 1,000 bananas from point B to the market M. Therefore, the distance between A and B must be chosen such that exactly 1,000 bananas arrive at point B. Now we can calculate the length of AB: 2000 - 3 × AB = 1000, so AB = 333⅓ kilometers. Note that this distance is less than 500 kilometers, so the camel can travel back from B to A. It follows that BM = 1000 - 200 - 333⅓ = 466²⁄₃ kilometers. As a result, the camel arrives at the market with 1000 - 466²⁄₃ = 533⅓ bananas.

The full scenario is as follows: first, the camel takes 1,000 bananas to point A. There, it drops 600 bananas and returns with 200 bananas. Then the camel takes another 1,000 bananas to point A. Again, it drops 600 bananas and returns with 200 bananas. After this, the camel takes the last 1,000 bananas from the plantation to point A. From point A, it leaves with 1,000 bananas to point B. At point B, it drops 333⅓ bananas and returns with 333⅓ bananas. Then it takes the second load of 1,000 bananas from point A to point B. Finally, it carries the remaining 1,000 bananas from point B to the market, where it arrives with 533⅓ bananas.

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