You can also convince yourself that if the corner of any unit square lies on , then its coordinates must add up to an even number. If such a point were also the corner of one of the rectangles, then our small square would be tiled by rectangles with sides of lengths and . Our path ends at the point that comes from repeatedly reflecting the top right corner of in sides of rectangles, so the end point of is a corner of . Have a look at the Geogebra animation below (the play button is in the bottom left corner) and try to figure out how the construction works. From we’ll now create a zig-zag path , which lies entirely within the bottom left rectangle , using repeated reflection. Conversely, the fact that every unit square is crossed by means that every point with coordinates that add to an even number lies on . It then follows that the self-intersection point closest to the starting point lies on the first segment of the path at distance from the starting corner and that is the number of unit squares crossed by the first segment of the path from the starting point to the closest point of self-intersection.

We claim that the ball eventually hits a corner and that the least common multiple of and is the length of the path the ball has traversed until it hit the corner, divided by . To prove that this is really the case, recall that the ball will make 90 degree turns whenever it hits a side of . You can force your opponent to make a penalty at some points. To know who will begin the match, you will have to string, which can either be based on an imaginary line (head string) or the number of wins (scoring string). There are three ways of scoring: (1) the losing hazard, or loser, is a stroke in which the striker’s cue ball is pocketed after contact with another ball; (2) the winning hazard, or pot, is a stroke in which a ball other than the striker’s cue ball is pocketed after contact with another ball; (3) the cannon, or carom, is a scoring sequence in which the striker’s cue ball contacts the two other balls successively or simultaneously. The skill involved consists of developing one scoring stroke after another. To do this, note that at least one of or must be odd, and that the end point of comes from repeatedly reflecting the top right corner of the rectangle .

If that were the case, we wouldn’t know which of the two sides that meet at the corner to reflect in. Given our two numbers and , none of which is a multiple of the other, start by forming a square with sides of length . But this is impossible: as we noted above, the square with side length is the smallest square that can be tiled in this way. Any unit square only has two corners whose coordinates add up to an even number and these are diagonally opposite each other. Coordinates of the corners of unit squares that lie on t are shown in red. 3. What are the symmetries of the arithmetic billiard path (as a geometrical figure)? 1. If one of the two given numbers is a multiple of the other, what is the shape of the arithmetic billiard path? To pocket the eight ball in a designated pocket, one must first pocket all their designated group of balls (either stripes vs. Furthermore, two points are awarded if a ball other than the player's cue ball strikes the pocket.

All three table sports are fun to play. A fluid stroke and precise aim are required to play pool like a pro. Basically, the point here is that modelling impacts like these is a tricky business. She would like to thank Andrew Bruce for help with the article. No matter how consistent you are with the first shot (the break), the smallest of differences in the speed and angle with which you strike the white ball will cause the pack of billiards to scatter in wildly different directions every time. It does not lose speed and, by the law of reflection, is reflected at a 45 degree angle each time it meets a side (thus the path only makes left or right 90 degree turns). Nature exhibits not simply a higher degree but an altogether different level of complexity. Play continues until only the six colours remain on the table. The billiard table consists of a flat playing surface with six pockets. The chapters in the book largely represent a faithful reconstruction of Astill’s playing career using contemporary newspaper reports and the writings of others as its foundation.

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