# The Football Puzzle Solution

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:
NS(N)
01
11
21
31
41
52
63
74
87

Now we can compute new values automatically:
NS(N)
87
910
1014
1121
1230
1344
1465
1595
16140
17205
18300
19440
20644
21944
221384
232028
242973
254357
266385
279358
2813714
2920099
3029457

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.

Last revised on 19 June 2005.