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### Team's Final Score

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Date: 10 Jul 1995 11:38:13 -0400
From: Anonymous

Question: The rules of a certain game allow a team to score either 3 points
or 8 points.  A team's final score will be any combination of these points.
For example,  14=3+3+8 and 27=8+8+8+3  Which numbers cannot be a team's
final score?

From: Michelle Arseneau
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```
Date: 11 Jul 1995 21:29:57 -0400
From: Dr. Ken

Hello there!

This is a great question.  To start out, we could draw a number line and
eliminate all the numbers that we know we _can_ get.  My number line here
will be a bunch of \$'s:

\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$... (there are 50
of them)
1234567...

First off, let's eliminate all the multiples of three:

\$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$ \$\$...

Now let's eliminate all the multiples of eight.  I'll put apostrophes in
them for clarity's sake:

\$\$ \$\$ \$' \$\$ \$\$ '\$ \$\$ \$\$ \$\$ \$\$ \$' \$\$ \$\$ '\$ \$\$ \$\$ \$\$...

Now notice something.  Everywhere there's an apostrophe, we can add threes
to that place value and eliminate more numbers.  I'll do it in two steps and
fill the slots with a , :

\$\$ \$\$ \$' \$, \$, ', \$, \$, \$, \$, \$' \$, \$, ', \$, \$, \$,
\$\$ \$\$ \$' \$, \$, ', ,, ,, ,, ,, ,' ,, ,, ', ,, ,, ,,

Hey!  There's not much left.  The first pass took out all the big numbers
that have a remainder of 2 when divided by 3, and the second took out all
the big numbers that have a remainder of 1 when divided by 3.  Since all the
numbers that have a remainder of 0 when divided by 3 were already gone,
there are no numbers past 13 left (all numbers have a remainder of 0, 1, or
2 when divided by 3).  So by looking at the diagram, we have our answer: we
could never get a score of 1,2,4,5,7,10, or 13:

\$\$ \$\$ \$' \$, \$, ', ,, ,, ,, ,, ,' ,, ,, ', ,, ,, ,,
12 45 7  10 13

Thanks for the question!

-K
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Associated Topics:
High School Discrete Mathematics

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