# Ghost Ship Collisions

There are five ghost pirate ships drifting on an infinite ocean in the spiritual realm. Each ship moves at a constant speed and never changes direction. No two ships are traveling in parallel paths.

These ships have been drifting for a very long time. In fact, time extends infinitely both to the future and the past, and these ships have drifted for all of history.

Because they are ghosts, two pirate ships can pass right through one another. However, such collisions release a huge amount of spiritual energy. For every pair of ships that collide, one trillion souls are destroyed. Since the there are only 10 trillion souls in the universe, these ships could destroy all life if every pair of them collided!

Eight pairs of ships have already collided: are the 2 trillion remaining souls doomed?

Think of the ships as point masses. If 4 ships meet at the same time, this would destroy 6 million souls, since 6 pairs of ships collided.

• You mean to say, there have been eight collisions. – Nefer007 Jun 23 '15 at 19:54
• Just to be clear, is this infinite ocean 2-dimensional? – Rand al'Thor Jun 23 '15 at 20:23
• Well, we can at least rule out the possibility that this is a spherical spiritual realm! (These collisions would be periodic in that case) – Milo Brandt Jun 24 '15 at 0:41
• The question cannot be answered without describing the shape of the ocean. Is it a toroidal (locally flat) ocean, if so what are the relative dimensions? How does 4 ships meeting result in 6 pairs of ships? Is it 4C2? – Aron Jun 24 '15 at 6:30

no, it is possible that the 2 trillion remaining souls will survive.

We can view the path of a pirate ship as a line in three-dimensional space (where the first two dimensions are spacial dimensions along the ocean, and the third dimension is time). Two pirate ships collide if and only if the corresponding lines intersect.

The given question is equivalent to:

Suppose we are given five straight lines in three-dimensional space, with no two lines parallel. Given that at least $8$ pairs of lines intersect, must all $10$ pairs of lines intersect?

The benefit of the equivalent question is that it is easier to visualize a counterexample. Consider a tetrahedron with one edge removed. This figure has five edges, which can be extended into five lines. We can count that exactly eight pairs of these lines intersect.

• Perfect answer :) I kind of messed up when asking this question, the problem is more interesting if you disallow three-or-more-way collisions. – Mike Earnest Jun 23 '15 at 20:41
• @MikeEarnest Before finding this solution, I worked out a "proof" that the remaining souls would perish. Only at the last minute did I realize I was assuming three way collisions weren't possible – Julian Rosen Jun 23 '15 at 21:09

Not very mathematical I know, but is this wrong?

No they're not doomed.

Because

time extends infinitely both to the future and the past, and these ships have drifted for all of history

They have already drifted for an infinite amount of time without colliding so it is already demonstrated—not theoretical—that they could continue to survive for an infinite future.

• This doesn't work; you can consider that the ships' paths up to that point to be rays in three dimensional space as Julian Rosen does. These rays intersect 8 times. The question asks whether, when the rays are extended to lines, they must intersect 10 times. – Milo Brandt Jun 24 '15 at 2:30
• @Meelo One answer is not incorrect simply because another is also correct. You're welcome to reduce the question to rays and lines if that helps you solve it, but the question remains "are they doomed?" nonetheless. If you want to rebuff this answer attack the logic of it. – Kanga Roo Jun 24 '15 at 2:48
• This logic does not work. Imagine the situtation with just two ships: just because they haven't collided yet doesn't mean they won't collide later. It could be the case that they will collide in 5 seconds. – Mike Earnest Jun 24 '15 at 3:20
• With a finite past no collision would indeed fail to guarantee they cannot not collide in the future but—the past here is infinite. Probabilistically aren't the odds of that collision occurring in the next 5 seconds or any point in the indefinite future, about one chance in infinity, ie: zero? – Kanga Roo Jun 24 '15 at 4:24
• I've just rolled a loaded dice an infinite number of times—and never once rolled a six. I'll give you a million dollars in exchange for your one dollar if I ever roll a six. How much do you bet? – Kanga Roo Jun 24 '15 at 5:06