Scoring System Problem

Date: 10/28/2001 at 00:47:48
From: Kate
Subject: Scoring system problem

A certain football league has the following scoring system: 

   - each field goal counts for 5 points        
   - each touchdown counts for 3 points

The only way to score points is with some combination of field goals 
and touchdowns. 

They have noticed that not every score is possible in this league 
(for example 1, 2, 4...); in fact they think that they know the 
highest score that is impossible to make. I have figured out that this 
is 7 points, but I would like to know why. How can I prove that this 
score is impossible and that all higher scores are possible?

Date: 10/30/2001 at 22:39:02
From: Doctor Douglas
Subject: Re: Scoring system problem

Hi Kate, and thanks for writing.  

Suppose in this example you demonstrate that it is possible to score 
8 points, 9 points, and 10 points (it is possible: 8 = 5 + 3, 
9 = 3 + 3 + 3, and 10 = 5 + 5). Now, we see that

You can make 11 by adding 3 to 8.
You can make 12 by adding 3 to 9.
You can make 13 by adding 3 to 10.
You can make 14 by adding 3 + 3 to 8.
You can make 15 by adding 3 + 3 to 9.
You can make 16 by adding 3 + 3 to 10.
You can make 17 by adding 3 + 3 + 3 to 8.
 .
 .
 .
and so on. You can make any higher score by following this pattern.

Thus, if you can find a set of consecutive scores where the size of 
the set is the smallest scoring unit (in this case this smallest
unit is 3 points), you can make all higher scores.  

Note that this doesn't prove that the score of 7 is indeed impossible.
For that you need the following type of reasoning:

If we can make 7 points, then either we can make 4 points (and add 3 
to it to make the total of 7), or we can make 2 points (and add 5 to 
it to make the total of 7).

Is a score of two possible? No, obviously not, since the smallest
scoring unit is 3. Is a score of 4 possible? If so, then either we can
make 1 point (and add 3 to it to make 4) or we can make -1 point (and 
add 5 to it to make 4). But we obviously cannot make 1 point or -1 
point, since the smallest scoring unit is 3.

So we conclude that we cannot make 4, and we have already concluded 
that we cannot make 2. So we cannot make 7.

I hope this helps answer your question why 7 is the largest impossible
score. If you need more help with this, please write back.

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