Planet Earth has been at war with the nefarious aliens of Planet XYZ for a while. The front lines are on Planet XYZ, and you, being Earth's lead scientists, need to find a way to get vital information to the front lines.
You have discovered a new type of spaceflight technology that allows you to reach Planet XYZ in $2$ days. This new spaceflight technology runs on Units of fuel. Each Unit a ship has in its fuel tank allows for $6$ hours of travel for that ship.
However, Planet XYZ has a laser shield around it that can only be bypassed by a sufficiently small ship, and the only ship you have that is that small can hold a maximum of $2$ Units of fuel, which is only enough for $1/4$ of the journey.
You also have $2$ fuel tank ships, that can run on Units of fuel at the same rate and travel at the same speed as your small ship, that can each hold $8$ Units. However, they can't bypass the shield. You have one low-quality fuel tank that may only transfer Units to a ship at the same spot as it. The other tank is high-quality and can transfer Units of fuel to another tank or the small ship if that ship is exactly $1/2$ or less of the total distance from Earth to Planet XYZ. Transferring Units removes a chosen whole number of Units from the tank of the transferring ship and adds them to the tank of the receiving ship. You can't bring a ship above its starting number of Units this way.
There is one more major complication; you've made a deal with Planet ABC (which is located $5/8$ of the way to Planet XYZ) to deliver both fuel tanks to them, that have at least $7$ Units combined, in which case, they will help you fight Planet XYZ.
A ship can refill on Earth up to its maximum number of Units. However, Earth only has $8$ Units of fuel remaining, besides the ones used to fill all of the ships at the start.
Refueling takes no time, as does transferring fuel, reversing direction, or landing and taking off from a planet.
Some final, minor rules.
- Ships that aren't on a planet must be in motion at all times, or Planet XYZ will blow them up. This includes ships that are out of fuel and aren't immediately refilled.
- You must deliver Planet ABC's payment at one time, not gradually. In addition, you can't take Units away from Planet ABC once you've given Units to them.
- Ships must be on Earth to refill from Earth's supply of Units.
- You only have 2 days before Planet XYZ unleashes their ultimate laser and wins the war. You must act immediately, and the main ship must be constantly moving forward. In addition, your deal with Planet ABC must be completed in this time.
How can you get your small ship to Planet XYZ and your tanks to Planet ABC containing at least $7$ units, both within 2 days?
Edit: Welp, I screwed up the question. I had come up with a solution beforehand without a Unit limit, and not glancing over it enough, I thought it only required a backup of 8 Units on Earth. Looking at it closer, it should be possible to do it with 16. My apologies.