How Far Can You Go?
Math brain teasers require computations to solve.
Your assignment is to make a delivery from the depot to your base camp, 1600 miles away. The trip begins normally, but unfortunately, halfway to base camp, your supply truck breaks down. You have no way to call for help. Luckily, in addition to medical supplies, the truck is carrying a Desert Patrol Vehicle (DPV) and two barrels of fuel.
The DPV has a full 10 gallon tank, but to make room for the medical supplies, it can only carry one of the 45 gallon barrels at a time. While you can't transfer fuel between barrels, you can refill the tank from the barrels. Assume the DPV can get 12 miles per gallon, regardless of the load it carries.
How far can you go? Is it far enough to deliver the medical supplies and the DPV to your base camp?
HintUse half of a tank to carry one barrel, then return for the other.
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Answer
You can travel 840 miles from the broken supply truck. Since your base camp is 800 miles away (half of 1600 miles), you can easily complete the delivery. Here's how:
Although you can only carry one barrel at a time, you need to transport both of them until one barrel is empty. F is the barrel used first, and L is the barrel used last.
Use half of a tank (5 gallons) to carry L 5*12=60 miles, and then return for F (uses another 5 gallons). Refill the tank from F and carry it to where you left L (also uses 5 gallons). Refill the tank again from F and switch barrels. Now you are 60 miles from the broken truck, the tank and L are full, but F is down to 30 gallons (45555=30). Repeat this procedure two times.
Now you are 60*3=180 miles from the broken truck. The tank and L are still full, but F is empty. Leave F behind and make a run for it, carrying L. Refill the tank from L as needed. There are 55 gallons of fuel left so you can travel an additional 55*12=660 miles, for a maximum of (180+660) 840 miles. Since you only have a total of 800 miles to go, you will make it to your base camp.
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Comments
monkeygrl99
Jul 13, 2006
 This wasn't hard, but wasn't easy either!!
Yaay! First one! 
dan1234
Jul 13, 2006
 Haha, ahhhhhh... "one at a time." 
scallio
Jul 20, 2006
 Very clever indeed. Great teaser!

Qrystal
Jul 20, 2006
 Ah, that was fun. Miscounted at the last moment and thought the med supplies were in trouble, but recounted and realized all was good. 
(user deleted)
Jul 22, 2006
 when you "Assume"... 
(user deleted)
Feb 02, 2007
 Lol I love the "assume" sentiment.
hard problem, at least for me. but its late and I'm tired, which probably didnt help .
Good job anyways 
lilredc5
May 24, 2007
 Good teaser. You know you would travel the same total distance but save yourself some time moving barrels around if you: begin and go a full 15 gallons worth of distance, leave your supplies there, go back for the second barrel. When you return with the full barrel you can drop off the empty one and pick up your supplies and travel the remaining 55 gallons worth (70 gallons of distance total = 840 miles) 
javaguru
Jan 02, 2009
 The answer was far easier to find than the explanation suggests.
100 gallons of fuel means that there is a 1200 mile range. The final sprint with one barrel and a full tank is 55x12 = 660 miles. Up until that point you have to cover the ground 3 times, so the 1200660=540 miles you travel before emptying the first barrel will take you 540/3 = 180 miles from the truck. 660+180 = 840.
I didn't even bother working out the specific trips to get to 180. 
nejco13
May 22, 2010
 How can you start by filling up the truck with a barrel F, when the tank is already full.
In order to do that you would have to somehow get that 10 gallons, that are already in tank, out. So there is no way to save that 10 gallons for the "final sprint". You have to use it at start. So when you use that 10 gallons at start, you can get to 40 miles, where you have both full barrels and medical stuff. So from then on you use one full barrel 3 times in order do cover the distance of 180 miles. So now you are at a distance of 220 miles from broken truck and one full barrel of gas. You can cover a distance of 540 miles with that barrel. So you can get to 760 miles from the broken truck. I cant get further than this? 
nejco13
May 22, 2010
 Ok, I Just figured it out. Haha
Nice one. 
opqpop
Sep 10, 2010
 @lilredc5
When you travel 15 gallons, you have to travel back 15 gallons. That is impossible if you are leaving a barrel and hence coming back with no barrels.
This is a great teaser. Kudos to the author. 
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