The entire second grade at Sunshine Elementary participated in a single elimination checkers tournament. There are a total of 118 second graders and so there were 59 matches in the first round of the tournament. In the second round,the 59 remaining students were paired, resulting in 29 matches: one student received a bye (that is, did not have to play in that round) In the third round, the 30 players remaining were then paired and so on.
a) How many matches in all were required to determine a winner of the tournament?(a match is a single game between two students)
b) How many matches would be required if n students participated in the tournament? Why?
a) To determine the total number of matches required to determine a winner of the tournament, we need to count the matches in each round and sum them up.
In the first round, there were 59 matches.
In the second round, there were 29 matches since one student received a bye.
Continuing this pattern, in the third round, there would be 14 matches, in the fourth round there would be 7 matches, and so on.
Adding up the matches from each round:
59 + 29 + 14 + 7 + ... + (final round with 1 match)
This is an example of a geometric series, where each term is half of the previous term.
To find the sum of the geometric series, we can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r),
where:
S is the sum of the series,
a is the first term (59 in this case),
r is the common ratio (1/2, since each term is half of the previous term),
n is the number of terms (in this case, we have n rounds).
If we substitute the values into the formula, we get:
S = 59 * (1 - (1/2)^n) / (1 - 1/2).
Since we want to find the total number of matches required to determine a winner, we want to find the sum of all the terms, including the final round with 1 match. This means we need to consider n rounds.
So, the total number of matches required to determine a winner is:
S = 59 * (1 - (1/2)^n) / (1 - 1/2).
b) If n students participated in the tournament, the number of matches required can be found using the same formula for the sum of a geometric series. However, we need to determine the number of rounds based on the given number of students.
We know that in each round, the number of matches is half of the previous round, except for one round where one student receives a bye.
To determine the number of rounds, we need to find the largest power of 2 that is less than or equal to n. This is because the number of matches in each round is halved, and each new round starts with the remaining students.
So, if n students participate in the tournament, the number of rounds would be log2(n) rounded down to the nearest whole number. The number of matches in each round can be calculated by dividing the number of matches in the previous round by 2, except for the round with the bye.
Using this information, we can apply the formula to find the number of matches required.
To summarize, the number of matches required if n students participate in the tournament can be found by determining the number of rounds (log2(n) rounded down to the nearest whole number), calculating the number of matches in each round (half of the previous round, except for the round with the bye), and summing up the matches from all rounds using the formula for the sum of a geometric series.