Can you solve the alien probe riddle? – Dan Finkel


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? Pause here if you want
to figure it out yourself. 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.