# The Dark Matter Fuel Riddle

It’s an incredible discovery: an abandoned alien space station filled with precursor technology. Now every species in the galaxy is in a mad dash to get there first, but you’ve got a problem. Your ship can’t hold enough fuel to get you there unless you vent caches of it into space at precise points and then come back for the fuel later. Can you reach the alien space station? Dan Finkel shows how.

### Transcript:

It’s an incredible discovery: an ancient, abandoned alien space station filled with precursor technology. But now, every species in the galaxy is in a mad dash to get there first and claim it for themselves.

And right away, you’ve got a problem. Your ship’s faster-than-light jump drive consumes 1 unit of fuel for every parsec of distance it takes you, and your ship holds only 15 units of fuel. But the space station is 23 parsecs away, and there’s only empty space between there and here.

There’s one thing that can help you, though: dark matter fuel is stable in deep space. That means you can vent a cache of it from your fuel chamber, and then come back to pick it up again later. Even though your ship holds only 15 units of fuel, you’ve been granted use of all 45 units in your current location. With some strategic fuel caching along the way, you might be able to make it all 23 parsecs.

So how can you reach the alien space station? Answer in 3, 2, 1.

It’s possible to solve this riddle using as few as two cache points, and there are also valid solutions that use more. No matter how you go about it though, the key is determining exactly where to cache fuel along your route.

Let’s work backwards from the alien space station. To reach 23 parsecs, you’ll have to leave the 8-parsec mark with a full tank of fuel. The 8-parsec point is too far from the start to use as a cache right away; you could jump there, but wouldn’t have enough fuel to return to the start, let alone store any for later. So that means you’ll need to find a cache somewhere between the start and 8. But where?

There’s an interesting pattern that can help. At the start you have exactly 3 tanks’ worth of fuel. At 8 parsecs you need exactly 1. Is there a point, which we can call point X, where you could have exactly 2? That would be useful, because then you could refuel there exactly twice, making full use of your storage capacity without any waste.

Wherever point X is, you’ll jump forward from it twice: once to deposit some fuel at the 8-parsec cache point, and a second time for good. So you’ll jump the distance between X and 8-parsecs 3 times in all. You’d have 2 tanks of fuel at point X, and need 1 left at the 8 parsec cache point, so you can spend one tank– or 15 units– going back and forth. Since 15 units divided by 3 trips is 5, we can place these two cache points 5 parsecs apart. Any farther, and you wouldn’t have enough fuel to reach the alien space station.

So it looks like the earliest we can place point X is at the 3-parsec mark. Is it possible to transport 30 units of fuel there? Let’s try.

You set out with a full tank of 15 units. You jump 3 parsecs, drop 9 units off at the cache point, and then jump the 3 units home, arriving with an empty tank. Repeating this process gets you 18 units of fuel at the cache point, and one more jump puts you at the 3-parsec cache with 30 total units of fuel. So far so good!

Next, you jump to the 8-parsec mark, drop off 5 units of fuel, and jump back to the 3-parsec mark. You fill up your tank and jump forward again, arriving with 10 units of fuel in your tank.

And now the end is in sight. You beam the 5 units of fuel in from deep space to fill your tank to capacity, and type in the coordinates of your final destination. A 15-parsec jump leaves you running on fumes, but ready to dock with the precursor space station.

Time to put this alien tech to work and make life better for everyone in the galaxy.

### Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.