# The Football Puzzle

After watching the football game, Stan runs into Ollie, who asks what the final score was.

Stan says "We won with 20 points."

Ollie says, "Why, that's a very common score!"

Stan says, "What do you mean?"

Ollie says, "Why, I mean there are so many ways to achieve it. You could have two touchdowns and extra points, plus one "bare" touchdown, for instance. You could have two touchdowns with two-point conversions, and two safeties. You could have four field goals plus a touchdown with two-point conversion. And of course, it's really necessary to record the order in which the individual scores were made. I haven't begun to list all the possibilities!"

"Well, Ollie, I don't think you can list all the possiblities, and I'm not going to wait here while you try!", said Stan.

Can you determine how many ways a score of 20 points can be achieved? Two ways are different if they are made up of different scores, or of the sames scores but in different order. Can you do it in a systematic way, which could easily be used to make a table of the number of ways of making any score?

Assume the following possibilities for scoring:

• If the other team doesn't show up, the game is forfeit, and you score 1 point (and the game is over.)
• On defense, if you tackle the other team's quarterback in their endzone, you get 2 points for a "safety".
• On offense, you can kick the ball over the defense's goalposts for a 3 point field goal.
• On offense, you can carry the ball across the defense's goalline for a 6 point touchdown.
• On offense, immediately after a touchdown, you can kick the ball through the defense's goalposts for an additional 1 point ("point after touchdown"). We can combine this with the touchdown and write it as a (6+1) or 7 point score;
• On offense, immediately after a touchdown, you can make another play to carry the ball across the defense's goalline for an additional 2 points ("two point conversion"). We can combine this with the touchdown and write it as a (6+2) or 8 point score;

I give up, show me the solution.

Last revised on 19 June 2005.