English allows you to alter the trajectory of the cue ball and bend or curve around other balls on the table. A snooker table, particularly the English variant, is quite massive. For pool tables and snooker tables, you will see six pockets. These three - billiards, pool, and snooker - all use cue sticks, tables, and balls. The balls are played in ascending numerical order and the lowest numbered ball must be contacted by the cue ball in order to establish a legal hit. Just make sure not to hit in the 8 Ball before all of your other balls are sunk, or else it is game over for you! This will make it easier to pocket the balls. His jagged table is made of 29 such triangles, arranged to make clever use of this fact. His approach worked not only for obtuse triangles, but for far more complicated shapes: Irregular 100-sided tables, say, or polygons whose walls zig and zag creating nooks and crannies, have periodic orbits, so long as the angles are rational.

As you might remember from high school geometry, there are several kinds of triangles: acute triangles, where all three internal angles are less than 90 degrees; right triangles, which have a 90-degree angle; and obtuse triangles, which have one angle that is more than 90 degrees. In 2018, Jacob Garber, Boyan Marinov, Kenneth Moore and George Tokarsky at the University of Alberta extended this threshold to 112.3 degrees. He is also a guest lecturer at Stanford University for a new course on building Facebook Apps. In 2014, Maryam Mirzakhani, a mathematician at Stanford University, became the first woman to win the Fields medal, math’s most prestigious award, what is billiards for her work on the moduli spaces of Riemann surfaces - a sort of generalization of the doughnuts that Masur used to show that all polygonal tables with rational angles have periodic orbits. Billiard tables shaped like acute and right triangles have periodic trajectories. But in 1995, Tokarsky used a simple fact about triangles to create a blockish 26-sided polygon with two points that are mutually inaccessible, shown below. In 2016, Samuel Lelièvre of Paris-Saclay University, Thierry Monteil of the French National Center for Scientific Research and Barak Weiss of Tel Aviv University applied a number of Mirzakhani’s results to show that any point in a rational polygon illuminates all points except finitely many.

In 2019 Amit Wolecki, then a graduate student at Tel Aviv University, applied this same technique to produce a shape with 22 sides (shown below). It’s unknown if a shape with fewer sides exists. Pool uses a rack that arranges the balls in a triangular shape when the game starts. The other balls are placed in the triangle without purposeful or intentional pattern. This inscribed triangle is a periodic billiard trajectory called the Fagnano orbit, named for Giovanni Fagnano, who in 1775 showed that this triangle has the smallest perimeter of all inscribed triangles. But obtuse triangles remain a mystery. In their 1992 paper, Galperin and his collaborators came up with a variety of methods of reflecting obtuse triangles in a way that lets you create periodic orbits, but the methods only worked for some special cases. In the early 1990s, Fred Holt at the University of Washington and Gregory Galperin and his collaborators at Moscow State University independently showed that every right triangle has periodic orbits. The basic skills of pin tumbler lock picking include selecting the proper tools, manipulating pins through the keyway, applying torque, and recognizing the state of each pin. In the locked state the plug is prevented from rotating by a set of movable pin stacks, typically under spring pressure, that protrude from holes in the top of the opening in the shell into corresponding holes drilled into the top of the plug.

The typical combination lock design involves a set of (usually three or four) disk tumblers around a spindle connected to the external dial. Raking, in contrast, is a class of picking techniques in which several pin stack may be set at the shear line simultaneously. While some of these features can be defeated with conventional picking tools and are covered here, picking high security locks generally requires specialized tools and techniques (often designed for a specific brand or model of lock) and are beyond our scope here. From the attacker's perspective, too, lock picking is rarely the most efficient, most economical, fastest, or easiest method of entry. We identified 10 categories, or skills, that go into athleticism, and then asked our eight panelists to assign a number from 1 to 10 to the demands each sport makes of each of those 10 skills. In fact, it’s often a question that we get asked.