A bunch of math teachers decided to create their own football league where they

Question

A bunch of math teachers decided to create their own football league where they changed the rules of the game some what. Each field goal counts 5 points and each touchdown counts for 3 points. Obviously they preferred kicking to running. They did not allow any other forms of scoring points. Since they were more thinkers than runners, one of the team members noticed that some scores were impossible to get. For example you cannot have a score of 1 or 2 or 4. The question that they got distracted by was whether there is a highest number that can NOT be a score in the game. Of course they sit on the bleachers and start to calculate.

1. Try some other scoring patterns and determine if they have scores that are impossible to achieve. Of these are there some scoring patterns that do NOT have a highest impossible score? See if you can find a rule that will let you know if there is or if there is not a highest impossible score.

2. For those scoring patterns where you do have a highest impossible score, see if you can figure out a formula to determine what that highest impossible score is.

3. How is this connected to least common multiples?

Explanation / Answer

1) 7 is maximum score which in not possible.

in fact if (m and n) are co-prime numbers

then maximum number which is not possible by combination of these are

mn-m-n

here m = 3 ,n = 5

so 3*5-3-5 = 7

clearly 7 is not possible by combination of 3 and 5 .

,now we will prove that

n>=8 is possible by using 3 and 5.

P(n) be statement that score of n can be formed using 3 and 5.

(a) Showing that the statements P(8), P(9), and P(10) are true, completing the basis step of the proof.

8 = 3 + 5, 9 = 3 + 3 + 3, 10 = 5 + 5, so that P(8),
P(9), and P(10) are true.

(b) inductive hypothesis of the proof

The inductive hypothesis is that P( n) is true for 8 n k,
where k 10. (Notice that this is a strong induction proof, which
requires a stronger hypothesis.)

(c) need to prove in the inductive step

We need to prove in the inductive step that P( k + 1) is true.

(d)Completing the inductive step for k 10

If k 10, then k + 1 = (k 2) + 3. Since k 2 8, by the

induction hypothesis we have that P(k 2) is true, i.e., a score of
k 2 can be scored by using 3 and 5. Adding one
3 , we can score k + 1 points , i.e ., P( k +1) is true.

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