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.