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(a) (i) There are now four teams A, B, C and D in the competition.
Write down the 6 different games played by these four teams.
(ii) There are now five teams A, B, C, D and E in the competition.
Write down all the different games played by these five teams.
(b) Complete the table.
Use your answer to part (a)(ii) to help you.
[Table_1]
\begin{array}{|c|c|c|c|c|c|c|} \hline \text{Number of teams (n)} & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Number of games (g)} & 1 & 3 & 6 & & & \\ \hline \end{array}
(c) (i) The numbers of games in the second row of the table form a sequence.
Write down the mathematical name for this sequence.
(ii) Write down the rule to find further terms in this sequence.
(d) When there are $n$ teams in the competition the number of games played is $g$, where
$$g = \frac{1}{2}n^2 + kn.$$
Find the value of $k$.
(e) Using your value of $k$, show that the formula in part (d) is correct for 8 teams.
(f) Find the number of games played when there are 20 teams in the competition.
551
512
584
There are now 8 teams in the competition, A, B, C, D, E, F, G and H. Every team plays one game each week. Note that AB is the same as BA.
(a) Complete the table to show the games for the first three weeks. There are many ways of doing this. You only need to show one way.
[Table_1]
(b) Write down the total number of weeks it will take to play all the games................................................................
(c) Points are awarded to each of the 8 teams in the competition every time they play one of their 7 games.
A team gets
• 3 points for a win
• 1 point for a draw (when the score for each team is the same)
• 0 points for a loss.
A team plays all 7 games.
(i) Find the highest number of points that this team can get.
...............................................................................
(ii) Write down the smallest number of points that this team can get.
....................................................................................
(iii) The team wins two games, loses two games and draws the rest.
Calculate the number of points that this team gets.
....................................................................................
(iv) Can this team finish the competition with 20 points?
Show how you decide.
(v) Find all the different ways that this team could score 9 points.
One way is 1 win and 6 draws.
507
542
528
