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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.

  1. 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.
  2. 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.
  3. Ships must be on Earth to refill from Earth's supply of Units.
  4. 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.

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I believe that this

Can't be done in the time allotted

The strategy that comes to mind

Is to have the Good tanker trail, remaining on Earth for as long as possible to send fuel to the other two ships.

After 18 hours

The small ship and bad tanker will both be fully fueled, containing 2 and 8 units respectively, 3/8 of the way to XYZ, the good tanker will also be fully fueled, containing 8 units, and Earth will have 2 units remaining. At this point, the good tanker is forced to leave Earth if it wants to reach ABC within the time limit.

At this point

The Small ship needs to travel 5 additional units, the lead tanker needs to travel 2 additional units, and the trailing tanker needs to travel 5 units, for a total of 12. The ships total only 18 units of fuel, leaving a maximum of 6 units of fuel in the tankers once they reach ABC.

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  • $\begingroup$ The flaw in your reasoning is the good tanker does not have to reach ABC. Instead it can fuel the bad tanker which delivers the fuel. $\endgroup$ – Bennett Bernardoni Sep 27 '18 at 18:12
  • $\begingroup$ @BennettBernardoni "...deliver both fuel tanks to them, that have at least 7 Units combined" $\endgroup$ – Sconibulus Sep 27 '18 at 18:14
  • $\begingroup$ I didn't see that part. In that case I agree with your answer. This can only be solved with dropping that requirement or 6 extra hours. $\endgroup$ – Bennett Bernardoni Sep 27 '18 at 18:19
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Here is the ship movement log:

Day 0 Hour 0: All ships fueled on earth. The small ship and low-quality fuel tank head off for planet ABC.
Day 0 Hour 12: The small ship needs a refuel from the high-quality ship which refuels from earth.
Day 1 Hour 0: The small ship and the low-quality ship refuel from the high-quality ship which refuels from earth. The high-quality ship leaves earth.
Day 1 Hour 6: The small ship and low-quality ship reach planet ABC. The small ship continues to XYZ and the low-quality ship delivers its 7 fuel and stays there.
Day 1 Hour 12: The small ship needs a refuel from the high-quality ship.
Day 2 Hour 0: The small ship reaches planet XYZ.

And here is the fuel log:

+--------+-----+-----+-----+-----+-----+
|Day:Hour|Small| Low | High|Earth| ABC |
+--------+-----+-----+-----+-----+-----+
|   0:00 |   2 |   8 |   8 |   8 |   0 |
|   0:06 |   1 |   7 |   8 |   8 |   0 |
|   0:12 |   2 |   6 |   8 |   6 |   0 |
|   0:18 |   1 |   5 |   8 |   6 |   0 |
|   1:00 |   2 |   8 |   8 |   0 |   0 |
|   1:06 |   1 |   0 |   7 |   0 |   7 |
|   1:12 |   2 |   0 |   4 |   0 |   7 |
|   1:18 |   1 |   0 |   3 |   0 |   7 |
|   2:00 |   0 |   0 |   2 |   0 |   7 |
+--------+-----+-----+-----+-----+-----+

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