Football Scoring

Date: 11/24/2002 at 11:43:45
From: M. Barnes
Subject: Advanced algebra problem

In a football game, the blue team scored in some manner 10 different 
times (touchdowns, points-after-touchdown or extra points, and field 
goals; no safeties were scored by either team).
  
Additional facts to consider: 

  - the blue team missed 2 points after touchdown
  - the blue team score was an odd, composite number
  - the blue team outscored the red team by 18 points  

1. What was the number of points scored by the red team?
2. What is the smallest number of scoring plays that the red team
   could have made to get its point total?

This game was played when points after touchdown only counted as 1 
point.

Date: 11/25/2002 at 09:45:38
From: Doctor Ian
Subject: Re: Advanced algebra problem

Hi,

This is an exercise in eliminating possibilities. For example, if you
score 10 times, what is the _largest_ possible score you can have? It
would be 10 touchdowns, or 60 points, right? And what is the 
_smallest_ score you can have? That would be 10 field goals, or 30
points, right?  

Once you know those things, you know that the score of the winning
team has to be in the range

  30 + 18 < blue score < 60
          -            -

This is a small enough set of possibilities that you can make a table:

  blue score     ways to get it in 10 scores    
     60          10 touchdowns
     59          
     58
     57          9 touchdowns, 1 field goal
     56
     55          9 touchdowns, 1 extra point
     54 
     53          

and so on. Some scores won't be possible.  

Once you have the possible scores for the blue team, you can do a
similar analysis for the red scores. (You know that the red score must 
be 18 points less than the blue score.)  Again, some scores won't be 
possible.  

As Sherlock Holmes said, when you've eliminated everything that's
impossible, whatever remains, however improbable, must be true. 

Is this enough to get started? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/