# The Alien Probe Riddle

Your team has developed a probe to study an alien monolith. It needs protective coatings — in red, purple or green — to cope with the environments it passes through. Can you figure out how to apply the colors so the probe survives the trip? Dan Finkel shows how.

### Transcript:

The discovery of an alien monolith on planet RH-1729 has scientists across the world racing to unlock its mysteries. Your engineering team has developed an elegant probe to study it. The probe is a collection of 27 cube modules capable of running all the scientific tests necessary to analyze the monolith. The modules can self-assemble into a large 3x3x3 cube, with each individual module placed anywhere in the cube, and at any orientation. It can also break itself apart and reassemble into any other orientation.

Now comes your job. The probe will need a special protective coating for each of the extreme environments it passes through. The red coating will seal it against the cold of deep space, the purple coating will protect it from the intense heat as it enters the atmosphere of RH-1729, and the green coating will shield it from the alien planet’s electric storms.

You can apply the coatings to each of the faces of all 27 of the cubic modules in any way you like, but each face can only take a single color coating. You need to figure out how you can apply the colors so the cubes can re-assemble themselves to show only red, then purple, then green.

How can you apply the colored coatings to the 27 cubes so the probe will be able to make the trip?

You can start by painting the outside of the complete cube red, since you’ll need that regardless. Then you can break it into 27 pieces, and look at what you have.

There are 8 corner cubes, which each have three red faces, 12 edge cubes, which have two red faces, 6 face cubes, which have 1 red face, and a single center cube, which has no red faces. You’ve painted a total of 54 faces red at this point, so you’ll need the same number of faces for the green and purple cubes, too. When you’re done, you’ll have painted 54 faces red, 54 faces green, and 54 faces purple. That’s 162 faces, which is precisely how many the cubes have in total. So there’s no margin for waste.

If there’s any way to do this, it’ll probably be highly symmetrical. Maybe you can use that to help you.

You look at the center cube. You’d better paint it half green and half purple, so you can use it as a corner for each of those cubes, and not waste a single face. There’ll need to be center cubes with no green and no purple too. So you take 2 corner cubes from the red cube and paint the 3 blank faces of 1 purple, and the 3 blank faces of the other green.

Now you’ve got the 6 face cubes that each have 1 face painted red. That leaves 5 empty faces on each. You can split them in half. In the first group, you paint 3 faces green and 2 faces purple; In the second group, paint 3 faces purple and 2 green. Counting on symmetry, you replicate these piles again with the colors rearranged. That gives you 6 with 1 green face, 6 with 1 red face, and 6 with 1 purple face.

Counting up what you’ve completely painted, you see 8 corner cubes in each color, 6 edge cubes in each color, 6 face cubes in each color, and 1 center cube. That means you just need 6 more edge cubes in green and purple. And there are exactly 6 cubes left, each with 4 empty faces. You paint 2 faces of each green and 2 faces of each purple.

And now you have a cube that’s perfectly painted to make an incredible trip. It rearranges itself to be red in deep space, purple as it enters RH-1729’s atmosphere, and green when it flies through the electric storms. As it reaches the monolith, you realize you’ve achieved something humans have dreamt of for eons: alien contact.

### Ali Kaya

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