As a coach : Apart from the basic calculation of scores, it is important for a coach to think A good example would be with the case of a basketball game. Here the coach has the responsibilities to place the fielders into the positions that they are very much capable of handling efficiently.
This decision occurs as a result of how good the player is able to move and how quick are his reflexes. Thus it is the velocity, a physical factor that contains a lot of mathematical computations, that gives an idea of forming appropriate strategies.
As a judge: The judge has various duties to be done for a game. In a soccer, the judge could be a scorer while the judge remains to be a judge in games such as wrestling. The common task is to calculate the scores and other factors. If there are 8 teams, what is an efficient way to schedule the matches that must take place? Another question, about which there is a huge literature but which will not be treated here, is how to decide on the winner based on the results or scores that the players attain.
For example, if one has 8 teams, could the number of wins of the eight teams in decreasing order be 6, 5, 5, 4, 4, 2, 2, 0? Questions about rankings for teams in tournaments are closely related to the issues of ranking candidates in an election or ranking choices for economic policy.
Graph theory, a branch of combinatorics which draws heavily on geometrical ideas, uses diagrams consisting of dots and lines to help get insight into a variety of mathematical problems. The complete graph on n vertices has exactly one edge between every pair of vertices.
In each case the vertices of the graph are labeled with the names of the people or teams involved in the "tournament" or competition. Think of the vertices dots of a complete graph as representing the teams in a tournament and think of an edge joining two teams as being a match played by those two teams.
Note that in the graph K n each vertex has n -1 edges at each vertex. The number of edges at a vertex of a graph is known as its degree or valence. Consider first the case where there are 4 teams that must play each other. These matches could be played in 6 time slots, say one a week for 6 weeks. However, it might be desirable if venues rooms; playing fields for the matches are available to have several matches per time slot and the games be completed over a shorter period of time.
When I use the phase "time slot," there are various possibilities as to how the matches are actually played. Note that two matches per time slot might mean that there would be two games at exactly the same time or that the games be played in the morning and afternoon on the same "court" of a single day. There are a variety of terms used other than time slots, and a common one is "rounds," which I will use interchangeably with time slots and Event Window. Figure 3 shows the details of how the scheduling could work.
Edges in the graph that have the same color would occur during one time slot. Thus, for Event Window 1 shown in blue there would be matches between team 0 and team 3 and team 1 and team 2; for Event Window 2 shown in black we would pair team 0 and team 1 and team 2 and team 3; and for Event Window 3 shown in red we would pair team 0 and team 2 and team 1 and team 3. In attempting to use the ideas above we come to a complication when we try to extend what we have done from 4 teams to 5 teams.
Since 5 is an odd number we can not merely have all the teams play in pairs during an Event Window. There is a natural way to handle this problem. The concept of a "bye" in sports scheduling refers to a team's not having to play a match game during a particular Event Window. If one has 5 teams, there are 10 matches games that must be carried out for a round robin tournament where each team plays every other.
Thus, in five Event Windows we can schedule the whole tournament. You can see the way a schedule for the five Event Windows can be constructed and see the team which has a bye in each Event Window by consulting Figure 4.
The edges in different colors signify which teams play in an Event Window. For example, the two yellow edges tell one can have teams 0 and 3 and 1 and 2 play each other in a single Event Window; for that event window team 4 would get a bye. The other pairings for each Event Window can be similarly handled. Note that there is considerable flexibility in the arrangement of the colors into the five Event Windows.
If a collection of edges are disjoint from each other it is called a matching. If a graph G has a matching M which includes all of the vertices of the graph, then M is said to be a perfect matching. A necessary condition for a perfect matching is that the number of vertices of the graph be even.
K 4 has a perfect matching while K 5 does not. However, it is not difficult to find examples, such as the one in Figure 5, which has an even number of vertices, every vertex of valence 3 e. A pioneer in using graph theory as a tool for solving scheduling problems has been Dominique De Werra , who has spent much of his career at the Polytechnic University of Lausanne. During that time he has made a variety of contributions to sports scheduling and operations research in general.
Many of the basic results were described during the 's by De Werra. Interest in this subject has grown so large that there is now an online discussion group devoted to sports scheduling issues from both practical and theoretical viewpoints.
Another name for a perfect matching is a 1-factor. A k -factor one is a subgraph of the graph which includes all the vertices of the graph and where each vertex in the subgraph has the valence degree k. So when a graph has a 1-factor, we can think of the vertices as teams and the edges as games which the vertices teams joined by an edge play against each other.
Returning to our sports scheduling situation, when we have a complete graph which has an even number of vertices, we can ask if it has a collection of 1-factors which include all of the edges of the graph. The coloring we found for K 4 in Figure 3 shows that this graph has a 1-factorization into three 1-factors. Because of the special way we drew K 4 it may not be clear that we can continue to find 1-factorizations of complete graphs with even numbers of vertices.
To see the different in suggestiveness of different drawings, look at this drawing of K 4 Figure 6. In this version we can see that the edges of different colors can be interpreted as being in "parallel classes. One can, in fact, interpret this diagram as a finite affine plane with 4 points. Every line of the plane has two points on it. There are six lines and 3 lines through every point.
Now we move up to round robin tournaments with 6 teams Figure 7. Fifteen matches are to be played. In light of what happened for four teams it is tempting to take a boundary edge 01 in Figure 7 of the regular hexagon shown, and construct a matching by using the edges that do not meet this edge that are parallel to it, as it were.
If we do this we get the games: 01, 25, and Proceeding around the boundary we get another two groups of matches: 12, 03, 45, and 23, 14, This seems to take us off to a good start. There are 6 edges remaining so our hope is to group these into two sets of size 3. However, unfortunately the six edges that remain form two disjoint triangles: edges 02, 24, 04 and 13, 15, Now since we can not pick two disjoint edges from either of these triangles we reach a dead end.
There is no way we can take our initial group of teams for the first three time windows and extend the result to two more time windows! Although mathematicians love to reason by analogy and try to apply simple principles to solve a problem at hand, sometimes the analogy may not hold up, as we see in this case.
However, we will not lose heart. Perhaps we can try some alternate systematic way to schedule 6 teams in a round robin tournament. The straight-away are still m each but the semicircular turns now have a radius that is 8 m more than the radius of the inside rail. Ten laps would be ten times that amount or m. An archery target consists of five concentric as shown.
The value for an arrow in each region starting from the inner circle is 9, 7, 5, 3, 1 points. In how many ways could five scoring arrows exam 29 points? Solution: We can set up Diophantine equations to model the problem. The examples in this section have shown you that math can be used in analyzing sport stats and scoring. Basketball is one of the world's most popular and widely viewed sports. In early December , Canadian American Dr. In , the Commonwealth Five, the first all-black professional team was founded.
The New York Renaissance was founded in When shooting a basketball you want the ball to hit the basket at as close to a right angle as possible. If we look at the graph of the range function we can get an idea of how hard the player must throw the ball in order to make a 3 point. After 50 games, he has made out of attempted free throws.
The offense attempts to score more runs than its opponents by hitting a ball thrown by the pitcher with a bat and moving counter-clockwise around a series of four bases: first, second, third and home plate. A run is scored when the runner advances around the bases and returns to home plate. By the early s, there were reports of a variety of un-codified bat-and-ball games recognizable as early forms of baseball being played around North America.
A professional baseball team has won 62 and lost 70 games. How many consecutive wins would bring them to the mark? How long is the throw from third base to first base on a professional baseball diamond where the bases are 90 feet apart? The winner was whoever hit the ball with the least number of strokes into a target several hundred yards away.
A golf-like game is, apocryphally, recorded as taking place on 26 February , in Leonean de-Vecht, where the Dutch played a game with a stick and leather ball was also played in 17th-century Netherlands and that this predates the game in Scotland. There are also other reports of earlier accounts of a golf-like game from continental Europe. Professional golfers frequently drive a golf ball or more yards, but very few have an average distance of or more yards for the year.
For example, at one point in a season, Tiger woods had an average of Solution: To find the average distance, find the total yards of all his drives divided by the number of drives. Place-kicker scored a school record of 17 points with field goals 3 points each and extra points 1 point each.
How many different ways could he have scored the 17 points? Solution: We could use a guessing process and determine the different ways to score 17 points, for example, 4 field goals and 5 extra points or 5 field goals and 2 extra points. However, if we are not careful in our analysis, we might miss some of the possible solutions.
Since each field goal is 3 points and each extra point is one point, we get the following. X 3 points Y 1 points 0 17 1 14 2 11 3 8 4 5 5 2 There are six ways in which the kicker could score 17 points. Now, the distance of the travel distance of the basketball can be found using the arc length equation:.
An example: If the average velocity of a basketball throw is 2. The above figure shows different angles and entry points of a basketball into a basketball hoop. The diameter of the hoop ring is 18 inches. As the basketball size is smaller than the hoop ring, there is always a constant hoop margin.
Hoop margin is the amount of space left in the hoop ring after the basketball enters it. Free throws, jump shots, and three-pointers enter at an angle that gives an oval entrance to the hoop. This changes the given hoop margin. Apparent hoop size is the apparent opening of the hoop to the ball.
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