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:
N | S(N) |
---|---|
0 | 1 |
1 | 1 |
2 | 1 |
3 | 1 |
4 | 1 |
5 | 2 |
6 | 3 |
7 | 4 |
8 | 7 |
Now we can compute new values automatically:
N | S(N) |
---|---|
8 | 7 |
9 | 10 |
10 | 14 |
11 | 21 |
12 | 30 |
13 | 44 |
14 | 65 |
15 | 95 |
16 | 140 |
17 | 205 |
18 | 300 |
19 | 440 |
20 | 644 |
21 | 944 |
22 | 1384 |
23 | 2028 |
24 | 2973 |
25 | 4357 |
26 | 6385 |
27 | 9358 |
28 | 13714 |
29 | 20099 |
30 | 29457 |
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.