A total of 129 players entered a single elimination handball tournament. In the first round of play, the top seeded player received a bye and the remaining 128 players played in 64 matches. Thus 65 players entered the second round of play. How many matches must be played to determine the tournament champion?
A carpenter has three large boxes. Inside each large box are two medium sized boxes. Inside each medium sized box are five small boxes. How many boxes are there altogether?
Hmm, the second one is quite simple, i will walk you through what you need to do:
There are 3 large boxes
There are 2 medium boxes in each large box.
Sense there are 3 large boxes and each box has 2, you could multiply 3x2 and get 6.
So far all together, there are 6 boxes.
In each meduim box there are 5 small boxes.
You could then multiply 6x5 and get a total of 30 boxes.
Hope this helped!
Well, it seems like the top seeded player was super lucky to get a bye in the first round. I guess they were just born with that extra teaspoon of skill!
Now, in the second round, we have 65 players left. To determine the tournament champion, we need to keep eliminating players until there's only one left, like something out of a magic trick!
Since there can only be one champion, each match in the second round will eliminate one player. So, with 65 players remaining, we need to play 64 matches to narrow it down to only 1 outstanding, victorious champion!
Phew, it sounds like a lot of handball action, but I'm sure it'll be a real nail-biter!
To determine the number of matches required to determine the tournament champion, we need to understand the structure of the tournament.
In the first round, the top seeded player received a bye, which means they automatically advanced to the second round without playing a match. The remaining 128 players played in 64 matches. So, after the first round, 65 players advanced to the second round.
In each subsequent round, the number of players is halved as players get eliminated. Therefore, to determine the number of matches required to determine the tournament champion, we need to figure out how many rounds are needed to reduce the number of players from 65 to 1.
Since each round reduces the number of players by half, we can represent this as a geometric sequence. The formula to find the number of terms in this sequence is given by:
n = log base 2 (65)
Using a calculator, the logarithm of 65 to the base 2 is approximately 6.02. However, since we cannot have a fraction of a match, we need to take the next highest whole number. Therefore, it would take a total of 7 rounds to determine the tournament champion.
Now, to calculate the number of matches required, we can sum up the number of matches played in each round. In the first round, there were 64 matches. In the second round, there were 32 matches, then 16 matches in the third round, 8 matches in the fourth round, 4 matches in the fifth round, 2 matches in the sixth round, and finally 1 match in the seventh round.
Adding these up, we have:
64 + 32 + 16 + 8 + 4 + 2 + 1 = 127
Therefore, a total of 127 matches must be played to determine the tournament champion.