Your internship in Professor Ramsey’s
physics lab has been amazing. Until, that is, the professor accidentally
stepped through a time portal. You’ve got just a minute to jump through
the portal to save him before it closes and leaves him stranded in history. Once you’re through it,
the portal will close, and your only way back will be
to create a new one using the chrono-nodules from your lab. Activated nodules connect to each other via red or blue tachyon entanglement. Activate more nodules and they’ll connect to all other nodules in the area. As soon as a red or blue triangle is
created with a nodule at each point, it opens a doorway through time that
will take you back to the present. But the color of each individual
connection manifests at random, and there’s no way to choose
or change its color. And there’s one more problem: each individual nodule creates a
temporal instability that raises the chances the portal
might collapse as you go through it. So the fewer you bring, the better. The portal’s about to close. What’s the minimum number of nodules
you need to bring to be certain you’ll create a red or
blue triangle and get back to the present? Pause here if you want to figure it out for yourself! Answer in: 3 Answer in: 2 Answer in: 1 This question is so rich that an entire
branch of mathematics known as Ramsey Theory developed from it. Ramsey Theory is home to some
famously difficult problems. This one isn’t easy, but it can be handled if you approach it systematically. Imagine you brought just three nodules. Would that be enough? No - for example,
you might have two blue and one red connection,
and be stuck in the past forever. Would four nodules be enough?
No - there are many arrangements here that don’t give a blue or red triangle. What about five? It turns out there is an arrangement of
connections that avoids creating
a blue or red triangle. These smaller triangles don’t count because
they don’t have a nodule at each corner. However, six nodules will always create a
blue triangle or a red triangle. Here’s how we can prove that without
sorting through every possible case. Imagine activating the sixth nodule, and consider how it might connect
to the other five. It could do so in one of six ways: with five red connections, five blue
connections, or some mix of red and blue. Notice that every possibility has at least
three connections of the same color coming from this nodule. Let’s look at just the nodules
on the other end of those same three color connections. If the connections were blue, then any additional blue connection between
those three would give us a blue triangle. So the only way we could get in trouble is if all the connections
between them were red. But those three red connections
would give us a red triangle. No matter what happens,
we’ll get a red or a blue triangle, and open our doorway. On the other hand, if the original three connections
were all red instead of blue, the same argument still works,
with all the colors flipped. In other words, no matter how the
connections are colored, six nodules will always create a red or
blue triangle and a doorway leading home. So you grab six nodules and jump through
the portal. You were hoping your internship would
give you valuable life experience. Turns out, that didn’t take much time.
