Four ghost ships - \(A\), \(B\), \(C\) and \(D\) - sail on a night so foggy that visibility is nearly zero. All ships sail in a straight line, changing neither speed or heading. They have been sailing this way for a long time, and all have different headings.
Sometime in the night \(A\) collides with \(B\), but they pass through each other since they are ghost ships. However as they pass \(A\)’s captain hear \(B\)’s exclaim that it was their third collision that night.
Later in the night \(A\) run into \(C\) and hears the same exclamation from \(C\)’s captain.
Will \(A\) hit \(D\)?
\(A\) will definetly hit \(D\).
Proof
Plot the path of the ships in 3 dimensions, with time as the third dimension.
The paths of ships \(B\) and \(C\) form a plane (since they are different headings).
The path of \(A\) is on the same plane since it intersects with both \(B\) and \(C\). Likewise for \(D\). Hence the paths of \(A\) and \(D\) are co-planar.
\(A\) and \(D\) are on different headings, hence their paths intersect. They’ve both been sailing for a long time, hence their paths don’t intersect in the path, so they must intersect in the future.