Ship Wrecked?

Let $A$ be one of the ships that we know has sunk. By considering relative motion, we might as well assume that $A$ is stationary throughout the motion.

Since $A$ sinks, it is struck by each of the other three ships $B,C,D$ . Thus each of $B,C,D$ follow paths which pass through the fixed point $A$ . Let us also assume that $B$ and $C$ collide, and that $B$ and $D$ collide (so that $B$ sinks). We want to know the fate of $C$ and $D$ .

The paths, relative to $A$ , of $B,C,D$ can be expressed as $\mathbf{B}(t) = (\beta - t)\mathbf{b} \hspace{1cm} \mathbf{C}(t) = (\gamma - t)\mathbf{c} \hspace{1cm} \mathbf{D}(t) = (\delta - t)\mathbf{d}$ for some fixed nonzero vectors $\mathbf{b},\mathbf{c},\mathbf{d}$ are fixed constants $\beta,\gamma,\delta$ . SInce $B$ collides with $C$ , there exists $\tau$ such that $\mathbf{B}(\tau) = \mathbf{C}(\tau)$ . This implies that either $\mathbf{b}$ and $\mathbf{c}$ are parallel, or else that $\mathbf{B}(\tau) = \mathbf{C}(\tau) = \mathbf{0}$ , in which case $\tau = \beta = \gamma$ . But this last condition would imply that $A,B,C$ all collide at the same time, which is not allowed. Thus we deduce that $\mathbf{b}$ and $\mathbf{c}$ are parallel. Similarly we deduce that $\mathbf{b}$ and $\mathbf{d}$ are parallel.

Thus $\mathbf{c}$ and $\mathbf{d}$ are parallel, and hence there exists some time $\rho$ for which $\mathbf{C}(\rho) = \mathbf{D}(\rho)$ , and hence $C$ and $D$ collide.

[This is not yet a complete solution. Read Mark's comments for the details.]

Hint: Consider the lines $L_1 = (x_i , y_i, t )$ which track the position of ship $i$ at time $t$ . (This is known as the worldline of an object, which is it's path through spacetime.)

Suppose ships 3 and 4 are left. What can you say about $L_1, L_2, L_3$ ?

Since a ship requires to collide 3 times to sink, each of the two ships that sank have collided thrice. But only 5 collisions have occurred. Hence the two ships must have collided with each other. Hence we have accounted for one collision. Since the ships move in straight lines they cannot collide with the same ship twice. Hence each of the ships that sank have to have had one collision each with the the other two ships. If the trajectories of the ships are lines $L_ {1}, L_ {2}, L_ {3}$ and $L_ {4}$ . Clearly the lines $L_ {1}$ and $L_ {2}$ are concurrent with each other and intersect both the other lines individually. Hence the other two lines have to intersect as well. Hence counting the collisions of the other two ships we can come to the conclusion that they must also sink.