How i did the hard badge:
day 1: Pick tires, transmission & engine level 3 each, rest goes to fuel.
day 2: Max out fuel and buy the gun with all the bullets you can afford.
day 3: Buy some more ammunition (but NOT the last 2 slots) and spend the rest of your money on the boost.
Use the whole boost at the edge of the last hill. Good luck!
Finally I did it in 5 years, conquering one country per turn. For this you need to concentrate on macedonian pikemen and upgrade them to the maximum as soon as possible. I didn't use any cavalry until the end ... only for the last country (Persia) i bought a couple of silver shield guards, because i already had 99 macedonian pikemen with all upgrades and didn't know how to spend my money elseway.
Really? As you progress it should take a bit more strategy and a lot less luck to succeed :)