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Project AMP Dr. Antonio Quesada – Director, Project AMP
The tournament problem
Activity 1.
A tennis club has organized a round robin tournament among the top ten players to decide who the number one player in the club is. That is, each of the top 6 players in the club will play everyone else in this group to decide who the best is.
a. Assuming that there is a winner after each player has played all the others, how many games took place? Explain your answer clear and coherently.
b. Can you think of a different way of solving the problem?
c. Can you generalize your solution? That is, can you readily find how many games will be needed if there were 100 top players?
d. Fill in the following table. Can you see a pattern to explain the total number of games needed each time a new player is added to the total?
Number of top players / 2 / 3 / 4 / 5 / 6 / 7 / 8Number of games needed
- Can you think of a practical or important application for this problem in real life? Explain.
Extension. John's father won 40 million dollars in the lotto last week. He plans to buy a fleet of small planes to service a number of cities so that: i) from any city anyone can rent a plane to go to any other city, and ii) initially, any pair of cities will have two planes assigned to service the route between them. If every plane costs $300,000 and he wants to keep at least 2 millions for operational expenses, what is the largest number of cities that his company can serve? How many planes he will need to buy?
Activity 2.
a. If the tennis club had a total of 513 players, would it be reasonable to have a similar kind of (round robin) tournament among all the players to find out who is their number one? Explain your answer.
b. If instead of using a round robin tournament among the 513 players, the club decides to have a tournament by elimination (once a player looses a game it is automatically eliminated), how many games will have to be played? Explain your answer. Can you think of a simpler solution?
The Pairing Problem
Activity 1.
Mr. Smith likes to group his fifth grade students in teams consisting of a boy and a girl. This year 2/3 of all of the boys are paired to 3/5 of all of the girls.
1. In Mr. Smith's fifth grade class, are there the same number of boys and girls, or do the boys outnumbered the girls, or is it the other way around?
2. What fraction of the entire fifth grade class is paired up?
Remark: Also known as "The Condominium Problem," this is a good problem to demonstrate the three approaches for learning: teamwork, teacher-centered, and the algorithmic approach.
Activity 2.
To find the solution of the previous activity, we used the fact that the groups (boys and girls) did not have common elements. Hence we knew that the total number of students in the class was the sum of all the boys and all the girls. However, in this activity you will work with groups that may or may not have common elements.
Confusing information
After surveying her sixth grade students, Ms. Miller's knows that 2/3 of the students like football, 3/5 like basketball, and 1/2 of the class watches at least two hours of TV daily. If we know that all the students in Ms. Miller sixth grade class fit in a bus with capacity for 50 people, how many students are there?
Hint:
a) Considering that "1/2" of the class watches at least two hours of TV daily, could the class consist of 21 students? What kind of number must the total be?
b) Similarly, if 2/3 of the students like football, what must be true about the total?
c) From your conclusions on a) and b) what must be true about the total?
d) Can we repeat this argument with the third fraction?
Project AMP A Quesada Director Project AMP