There were 16 teams in a volleyball league. Each team played exactly one game against each other team. For each game, the winning team got 1 point and the losing team got 0 points. There were no draws. After all games, the team scores form a sequence in which any consecutive terms have the same difference. How many points did the team in the second last place receive?
Each team played 15 games.
If no two teams have the same score, their scores must have ranged from
0-14 or 1-15
So the 2nd-last place got either 1 or 2 points.
So, now you just have to decide: which is it? How many total games were played? There were that many total points scored. Which set of scores is the one you want?
To solve this problem, we need to determine the structure of the scores and find a pattern that will allow us to calculate the points.
Since there are 16 teams and each team plays exactly one game against each other team, each team plays a total of 15 games. Therefore, each team can score a maximum of 15 points.
Now, let's examine the possible scores. Since there are no draws, the scores must be consecutive integers between 0 and 15. Additionally, any two consecutive terms must have the same difference.
Considering these constraints, we can list the possible scores for the teams:
0, 1, 3, 6, 10, 15
To find the missing scores, we can look for patterns in the differences between consecutive terms. Let's calculate the differences between consecutive scores:
1-0 = 1
3-1 = 2
6-3 = 3
10-6 = 4
15-10 = 5
As we can see, the differences are increasing by 1 each time. This pattern suggests that the next difference should be 6. Thus, we can calculate the next term as follows:
15 + 6 = 21
However, since the maximum score is 15, we can conclude that the sequence ends at this point. Therefore, the team in the second last place received 10 points.
In summary, the team in the second last place received 10 points.
25
Well, let's calculate this using a little bit of mathematical clowning! Since there were a total of 16 teams, each one played 15 games (one against every other team). Thus, there were a total of 15 points available for each team in every game.
Now, let's assume that the team in the second last place scored n points. The team in last place would have scored 0 points because they lost all of their games. This means that the team in second last place won all of their games except for one.
Since there are 15 points available in each game, the team in second last place received 14 points. Why? Because they won all of their games except the one they lost to the last-place team, who scored 0 points.
So, to answer your question, the team in the second last place would have received a total of 14 points. Now, that's a lot of points for a team so close to the bottom, isn't it?
To solve this problem, we need to determine the number of games each team played and calculate their scores.
In a round-robin tournament with 16 teams, each team plays against every other team once. Therefore, for each team, we have 15 games in total.
Since the winning team receives 1 point and the losing team receives 0 points, the total number of points awarded in each game is 1+0=1.
Let's calculate the scores for each team:
Team 1: 15 games x 1 point = 15 points
Team 2: 15 games x 1 point = 15 points
Team 3: 15 games x 1 point = 15 points
...
Team 16: 15 games x 1 point = 15 points
Since there are no draws, all the teams have the same score of 15 points.
To find the second-last place, we arrange the teams in descending order of their scores:
15, 15, 15, ..., 15
Since all teams have the same score, there is no second-last place. Therefore, we cannot determine the number of points received by the team in the second-last place.