# Largest impossible American football scores

- Author:
- tyler.seacrest

In a variant* of American football, a team can either score a = 3 points (a field goal) or b = 7 points (a touchdown). Under these rules, some scores are impossible. Question: what is the largest impossible score?
For example, a score of 5 is impossible. If you score any touchdowns, your score is greater than 5, so to achieve 5 there must be no touchdowns. However, if you only score field goals, your score is 0, 3, 6, 9, etc. 5 is skipped over. So 5 is impossible. We are asking here whether there are any scores higher than 5 which are also impossible, and if so what is the highest impossible score.
In the GeoGebra file below, the score is calculated when X field goals are scored and Y touchdowns are scored (by one team). You can also change the values a and b to different values.
*We are assuming there are no safeties, and teams always score exactly 7 points for a touchdown, so no missed extra points or 2-point conversions.