Nevermind, I don't quite agree with your answer. I agree that the runner should have 2 full gallons of water when he is at 150th mile, but I don't agree that the number of gallons at 125th mile should be 4. Here's why: If I carry two gallons from 125th mile to 150th mile, then I will finish one gallon just going. I will need another gallon to come back. So, if I have that interval of 25 miles, then I will have to use one gallon while going and the other gallon while coming back--this does not save me any water. So, instead, the travelling interval should be 12.5 miles NOT 25 miles. In effect, the number of gallons of water I will require at different intervals now become: 150m: 2 (2^1) 137.5m: 4 (2^2) 125m: 8 (2^3) - - - 25m: 2^11 12.5m: 2^12 = 4096 That means I need 4096 gallons of water at a distance of 12.5 miles from the start. I am just trying to know the lowest number of trips that I have to make to cover 200 miles, so I do NOT need not to know how many gallons I need at 0th mile (as opposed to what you have said above). And to have 4096 gallons of water at 0th mile, I need to get back to start 4095 (4096-1) times. So, I think the correct answer should be 4095 times. Let me know if you do not agree.