The Desert Crossing

bin Hassad, an ancient Arabic mathematician, had to get to the Great Library at Alexandria in the shortest possible way to deliver a very important discovery in geometry. In his path at one stage was a desert of about 800 miles. He was a poor man, only able to afford to hire one camel for the crossing.

bin Hassad realised that no camel he could afford could go 800 miles without finding extra water in the desert. Of course, he would carry water and the camel would have its own reserves, but still, the limit of travel without finding water was 500 miles. The sum of what is in the camel plus any water in bags being carried is 500 miles worth. There was no limit to the availability of water (for the camel to drink or to carry in bags) at the starting end of the crossing.

Clearly, he would have to make a few trips into the desert to set up dumps of water. But a camel was hired on the basis of the distance it travelled, the measure being the amount of water taken. Camel owners judged how far their camels had travelled by noting the amount of water taken. The question was what was the least distance that could be travelled to get to the camel station on the other side of the desert, or, in this case, the least amount of water needed.

He asked the camel hire staff what they thought would be the best way to proceed. "Oh yes", they said, "the best way is to not worry the head at all because this can harm the mind that Allah gave us. Just fill up the camel and these water bags and leave water wherever the camel makes a coughing sound or whenever you feel tired. Come back here and get more water and keep doing this till anyone can see that there is enough water in the desert to cross with a camel. This is the best way"

"Yes!", thought bin Hassad, "Of course! They get more shekels the more their camels travel. But I, a poor man and a mathematician, must work out what the least number of miles are needed to cross".

He concocted a plan and crossed the desert and paid all his dues. His feat was the talk of all camel hiring stations in the Arabian Kingdom.

What was the (shortest) distance he ended up travelling by camel?