Charlie moves toward the Round's final resting place.

                 CHARLIE
          It's a reverse trajectory problem.
          Unknown source, unknown initial
          velocity. Countless exterior
          variables... It's doable.

He squats low by the bullet hole, looks back, reverse-tracing
the Round's path, through the hole in the opposite wall,
across the kitchen to the exterior hole, a spot of daylight.

A projectile in motion follows a parabolic trajectory. This path is the result of a constant downward acceleration due to gravity, with no horizontal acceleration (neglecting air resistance). The initial horizontal velocity determines the time and distance the projectile will travel.

                 CHARLIE
          Okay, so September's out... unless
          you want to get married at the
          conference.
            (being clever, jokes)
          We could march up the aisle during
          the keynote address. Everyone
          could toast us good luck with the
          Gale Shapley Algorithm...

Amita looks up, barely smiles. OFF Charlie's look --

                 AMITA
          Used in combinatorics to solve the
          Stable Marriage Problem. I get
          it... It just doesn't help much...

A set of marriages is unstable if a man and woman within the set could improve their happiness by marrying each other rather than staying with their current partners. This Demonstration shows the set of stable marriages that results from having the men rank females on the basis of a distance measure from themselves to each female. Reciprocally, the females rank the males on the basis of a distance measure from themselves to each male.

                 LIZ
          So how do we find our two containers?

                 CHARLIE
          4-D mapping. We graph the
          movements in three dimensions --

                 OTTO (O.S.)
          Over a fourth dimension - time.
          Charlie spins to see OTTO has entered, unannounced.

In 4D, multiplication by quaternions gives a linear transformation, which ensures that the branches have the same lengths relative to their trunk, and the same angles, regardless of the orientation of their trunk. The same thing happens in 2D with complex multiplication, but in 3D such a transformation is not possible, which is not unrelated to the forced vanishing of vector fields on the sphere.

Charlie works at the board. Amita's nearby, working on her
computer. Both continually refer to their color-coded
Calendar. There's a very friendly sense of competition.

                 CHARLIE
          You're treating it as a
          combinatorial optimization problem?

                 AMITA
          Go with what you know, right?
          What's wrong with that?

                 CHARLIE
          Nothing... if limitations on dates
          were discrete, but they're not.
          For instance, social convention
          prohibits marrying on Tuesdays...

Alan ENTERS as Amita glances at Charlie's boards.

                 AMITA
          You think you're going to have
          better luck with the Pigeonhole
          Principle? You're likely to get a
          date five years from now...

They both go back to their work.

In 1834, Johann Dirichlet noted that if there are five objects in four drawers then there is a drawer with two or more objects. The Schubfachprinzip, or drawer principle, got renamed as the pigeonhole principle, and became a powerful tool in mathematical proofs.