There is obviously only one way to get 0 points;
There is one way to get 1 point.
There is one way to get 2 points
There is one way to get 3 points;
There is one way to get 4 points (two safeties).
When we get to 5 points, we see that this can in two ways, by a safety and a field goal in either order.
For 6 points, we could get 3 safeties, or two field goals, or a touchdown.
We can get 7 points by 7, or 2+2+3 or 2+3+2 or 3+2+2.
We can get 8 points by 2+2+2+2, or ( 2+3+3, 3+2+3, 3+3+2 ), or ( 6+2, 2+6 ) or 8.
This is clearly getting out of hand! Let S(N) be the number of ways of achieving a score of N. Then we should see that, for N greater than 1, we have
S(N) = S(N-2) + S(N-3) + S(N-6) + S(N-7) + S(N-8)That is, to get a total score of N, your last score was a 2, 3, 6, 6+1 or 6+2. And the number of ways of getting a score of N is the sum of the number of ways of getting a score that is N-2, N-3, N-6, N-7 or N-8.
Our table begins like this:
Now we can compute new values automatically:
No wonder Stan didn't want to wait for Ollie to think of all the possibilities for 20 points!
FOOTBALL_WAYS.F90 is a simple FORTRAN program for computing the number of ways of making a given score in football.
FOOTBALL_WAYS.OUT is the output from the program for scores from 0 to 100.
Back to The Football Puzzle.