Please Solve

Question

Good Morning,

 

Please help me  to solve this problem

 

In order to select capable contestants for their quiz shows, some producers require prospective contestants to compete off camera. One show, “Fractured funny bones.” Usually chooses four candidates and conduct a series of head to head matches between each pair of them. The candidate who performs the best is then chosen to appear on the show.

The game is played as follows: the contestants are shown a key word, and whoever first thinks of a joke using that word in the punch line gets one point. The first player to get five points wins.

When Manny Morris, Nancy Novokov, Patti proctor and Thomas Twickenham competed against each other, the results were as follows:

Patti won all three of her games and her opponents scored a total of only 2 points against her.

Nancy scored a total of 10 points and her opponents also scored 10.

Manny and Thomas both scored 8 points but Manny allowed his opponents 15 whereas Tom’s opponents scored only 14. In addition, Tom scored more points against Patti than many did.

What was the outcome and the score of each head to head match?

Answer 1

Let's start off with the number of matches. we have a total of 6: MN, MP, MT, NP, NT, & PT

 

P(atti) won all of hers, so P = 5 in all cases, and 15 total

N(ancy) = 10 total

M(anny) = 8 total

T(homas) = 8 total

 

NP + NM + NT = 10 (for opponents)

 5  +  a   + b   = 10

 

MP + MN + MT = 15 (for opponents)

 5   + c   +  d   = 15

Since there's five points max per round, Manny lost each of them:

5    +  5  +  5 

 

TP + TN + TM = 14 (for opponents)

 5  +  e   + f  =  14

one short from 15, and we know from the previous hint that Manny lost, so

 5  +  5   + 4  = 14

 

Here's the standings that we know so far, then:

 

MN (? 5), MP (? 5), MT (4 5), NP (? 5), NT(5 ?), & PT (5 ?)

P = 15, 15 accounted for

M = 8, 4 accounted for; 4 missing

N = 10, 10 accounted for

T = 8, 5 accounted for; 3 missing

 

With Nancy's 10 points fully accounted for, we can make NP (0 5) (since 10 + 0 = 10):

MN (? 5), MP (? 5), MT (4 5), NP (0 5), NT(5 ?), & PT (5 ?)

M has 4 points left, and T three points.

 

Let's take a closer look at Patti's matches now.

We know N has 0, and T > M (Thomas scoring more) , so we have:

0 + T + M = 2.

Since T is more, not equal, T must be 2, so 0 + 2+ 0 = 2

 

Now we can fill in the rest of Patti's matches:

MN (? 5), MP (0 5), MT (4 5), NP (0 5), NT(5 ?), & PT (5 2)

P = 15, 15 accounted for
M = 8, 4 accounted for; 4 missing
N = 10, 10 accounted for
T = 8, 7 accounted for; 1 missing

 

Manny and Thomas only has one match left each unaccounted for, so we just put in what amount of points are left over, so now we have the final head-to-head scores:

MN   (4 5)

MP    (0 5)

MT    (4 5)

NP    (0 5)

NT    (5 1)

PT    (5 2)