This is another well known problem. Solutions have appeared in 'American Mathematical Monthly', 'Mathematical Gazette' and other places. My solution is abridged from one of Martin Gardner's books.
Call 500 miles a 'unit' and the petrol needed to go 500 miles a 'load'.
Two loads will take the truck a maximum of 1 and 1/3 units in 4 trips. Out 1/3 units, dump 1/3 load and return. Then out, pick up 1/3 of a load at the dump and drive one unit.
Three loads will take the truck 1 plus 1/3 plus 1/5 units.
Out 1/5 units, dump 3/5 and return.
Out to 1/5, pick up 1/5 ang go to 1/3, dump 1/3 and return picking up 1/3 and 1/5 units on the way.
Out to 1/5, pick up 1/5 and go to 1/3 pick up 1/3 and drive 1 unit.
This is not quite enough. Proceeding on similar lines, 4 loads, with dumps at 1/7, 1/5 and 1/3 units will reach 1+1/3+1/5+1/7 units which is just over 1.6 units (800 miles).
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