Football ScoringDate: 11/24/2002 at 11:43:45 From: M. Barnes Subject: Advanced algebra problem In a football game, the blue team scored in some manner 10 different times (touchdowns, points-after-touchdown or extra points, and field goals; no safeties were scored by either team). Additional facts to consider: - the blue team missed 2 points after touchdown - the blue team score was an odd, composite number - the blue team outscored the red team by 18 points 1. What was the number of points scored by the red team? 2. What is the smallest number of scoring plays that the red team could have made to get its point total? This game was played when points after touchdown only counted as 1 point. Date: 11/25/2002 at 09:45:38 From: Doctor Ian Subject: Re: Advanced algebra problem Hi, This is an exercise in eliminating possibilities. For example, if you score 10 times, what is the _largest_ possible score you can have? It would be 10 touchdowns, or 60 points, right? And what is the _smallest_ score you can have? That would be 10 field goals, or 30 points, right? Once you know those things, you know that the score of the winning team has to be in the range 30 + 18 < blue score < 60 - - This is a small enough set of possibilities that you can make a table: blue score ways to get it in 10 scores 60 10 touchdowns 59 58 57 9 touchdowns, 1 field goal 56 55 9 touchdowns, 1 extra point 54 53 and so on. Some scores won't be possible. Once you have the possible scores for the blue team, you can do a similar analysis for the red scores. (You know that the red score must be 18 points less than the blue score.) Again, some scores won't be possible. As Sherlock Holmes said, when you've eliminated everything that's impossible, whatever remains, however improbable, must be true. Is this enough to get started? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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