Your truck, carrying 10,000 lbs of sugar and a mule, breaks down.

You must use the mule to carry the sugar the rest of the way, a 1,000 mile stretch of straight road, to your home. You can load sugar on to the mule to carry, but there are two caveats: a) the mule can’t carry more than 1,000 lbs at any given time; and b) the mule will continuously eat the sugar he’s carrying, at the rate of 1 lb per mile moved, whenever he’s moving, and he will not move, in any direction, if he has no sugar to eat.

Assuming that sugar may only be transported by the mule (you can’t carry any, the mule can’t pull the truck, and you get no external help), and that you can’t prevent the mule from eating the sugar, what is the most amount of the sugar, if any, that can make it to your house?Find the smallest natural number that can be expressed as the sum of the squares of two, not necessarily distinct, natural numbers in two different ways.