Question: One of several differences between the Canadian and American games of football is that in Canada, a team can score a single point without first having scored a touchdown. So it is clear that any score is possible in the Canadian game. Is this so in the American game? Indeed this seems to be the case, even assuming (this is not true!) that in the United States, points can be scored only three at a time (with a field goal) or seven at a time (with a converted touchdown). Here is an argument.
Assume that k points can be achieved with multiples of 3 or 7. Here is how to reach k+ 1 points. If k points are achieved with at least two field goals, subtracting these and adding a touchdown gives k + I points. On the other hand, if the k points are achieved with at least two touchdowns, subtracting these and adding five field goals also gives k + I points. Does this argument show that any score is possible in American football? Can it be used to show something about the nature of possible scores?