POW Free Think Football

Kick it POW

Problem Statement: The Free Thinkers football league does things differently. They score their games differently than everyone else. Each field goal is worth 5 points and each touchdown is worth 3 points. The only way to score points in their league is with field goals, touchdowns, or some combination of field goals and touchdowns. One of the Free Thinkers has noticed that not every score is possible in their league. For example, a score of 1 point isn’t possible and neither are 2 or 4. She thinks that beyond a certain number, all scores are possible.

Numbers
Possible?
1
No
2
No
3
3
4
No
5
5
6
3+3
7
No
8
5+3
9
3+3+3
10
5+5
11
5+3+3
12
3+3+3+3
13
5+5+3
14
5+3+3+3
Process: The first thing that I did is list the numbers 1-50 on a sheet of paper. I started at 1 and crossed off each number that were possible and circled the numbers that weren’t possible. It is impossible to get 1, 2, 4, and 7. It is possible to get 3, 5, and 8-50. So then I listed the number 51-100. I used the same method as I did before. It was possible to get every number from 51-100. Then I created a table to have all of the information together that shows how each number can be possible.

Numbers
Possible?
15
5+5+5
16
5+5+3+3
17
5+3+3+3+3
18
5+5+5+3
19
5+5+3+3+3
20
5+5+5+5
21
5+5+5+3+3
22
5+5+3+3+3+3
23
5+5+5+5+3
24
5+5+5+3+3+3
25
5+5+5+5+5
26
5+5+5+5+3+3
27
5+5+5+3+3+3+3
28
5+5+5+5+5+3
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After I did that I had to come up with four other scoring systems. So the first scoring system that I came up with was 4 and 8. I wanted to see how the results of two even numbers would come out. So I used the same method as I did before and I also created a table.
Numbers
Possible?
Numbers
Possible?
Numbers
Possible?
Numbers
Possible?
1
No
8
8
15
No
22
No
2
No
9
No
16
4+4+4+4
23
No
3
No
10
No
17
No
24
8+4+4+4+4
4
4
11
No
18
No
25
No
5
No
12
8+4
19
No
26
No
6
No