# Math Help - [SOLVED] A really hard puzzle involving algebra

1. ## [SOLVED] A really hard puzzle involving algebra

I don't understand the second part:
here is the problem. I have written what I don't understand at the bottom of the problem, I was unable to copy it porperly so sorry for the capital letters:

Nikki, Mikey, Akeem, Katie and Davey have all entered a mathematics competition. In each round of the competition, the entrants were put into pairs as far as possible (if there were an odd number of participants, all but one would be paired) and answered a set of questions at the same time as their opponent.

The winner was the entrant who gave correct answers to most of the questions. The winner got two points, one point was awarded for a draw and zero points for a loss. Every entrant played another entrant exactly once.

As we know, Nikki, Mikey, Akeem, Katie and Davey are all very good mathematicians,and they were delighted to find that they had all won the competition with the same score. In fact they had all dropped only 4 points. All of the other entrants only scored 4 points.

When the judges examined the results, there had been no draws.

How many points did Nikki, Mikey, Akeem, Katie and Davey all score?

When they met their friends Sophie, Stevie, Sabina, Marge and Chrissie, they discovered that they had been in a different group of entrants for the same competition. The friends admitted that they had dropped four times as many points even though they all scored the same number, but they noted that in their round the other entrants also scored exactly the same number of points as they had each dropped.

Davey thought that there were the same number of entrants in each round, so he said that he was sorry that his friends were not through to the next round, but, Sophie said that they too got through.

How many entrants were in their round?
Suppose there are n other people. The total number of points scored between them is n² - n so the total including the friends is n ² - n + 60 and this equals to 16n.

n ² - n + 60 = 16n
n ² - 17n + 60 = 0
Factorise: ( n – 5 ) ( n - 12) = 0
n = 5 or n = 12

However, we know that n > 5, so n = 17
Total entrants: 17

2. Here is alternative solution.

If n is the total no. of entrants, then

points scored by friends= $(2n-18).5$

points scored by others= $(n-5)16$

since there were no draws, total points that are scored in matches= $n^2-n$

$(2n-18).5+(n-5)16=n^2-n$

which gives n=10,17

Still working on your friend's solution...

3. Here is explanation to the solution you provided.

Sophie, Stevie, Sabina, Marge and Chrissie are five friends....and there are n more people in there round.

(When I say "n" people from now on, they mean the people other than the above 5 friends)

the other entrants also scored exactly the same number of points as they had each dropped.
So the total points scored by n people= $16n$

Now divide all the matches held into three categories:
(1) Held between five friends
(2) Held between n people
(3) Held between n people and five friends

in case (2), the points scored by n people= $n^2-n$

in case (1), the points dropped by friends=20......hence points dropped by friends in case(3) =60

so, the points gained by n people in third case= $60$

total points gained by n people in three cases= $n^2-n+60$