Problem statement: The Free thinkers Football League is able to score 5 points for each field goal and 3 points for every touchdown. The football league can score with a combination of a field goal and a touchdown, or just one event at a time (one field goal or one touchdown). One team member on the Free Thinkers Football League notices that not every score is possible. For example, a score of 1 or 2 is not possible because a touchdown yields 3 points while a field goal yields 3 points. The Free Thinkers Football League wants to know if there is a score that represents the highest impossible score they won’t be able to get. Is there a highest impossible score?
Process:
As i mixed the different combinations of field goals and touchdowns in which i went to 31 i got all the number except for 1,2,4, and 7 that's how i have figured out that the highest impossible score was 17 because 5+5+5=15 you can't add 3 or you would get 18 so I went back i thought what plus what equals 7 and I got 3+4=7 but that wouldn't work. So I thought again 2+5=7 so i said what has a two so i had 5 and get 17 so i thought again what is 17-5=12 how can i get 12 so i took 12 and divided it by 5 it didn't work so i did it by 3 and got 4 so 3+3+3+3+5=17. Once you got past 17 you could get all the other including 17. It was the same to get the numbers that were in the 20 and 30’s just add can.
Solution: After creating the chart, I stopped at the combination of 30. I stopped here because I realized that when you look at an increment of 10 that is possible, let’s say from 20-30, I concluded that you can add 10 to each number in the increment to make the next set of 10. You can keep repeating this process to get every single other whole numerical possibility. I chose to add 10 because 10 is a possible point combination for this constraint since 10 is composed of two field goals, or in equation terms 5+5=10. For example: 1. 20+10= 30 (30 is composed of 6 field goals) 2. 21+10= 31 (31 can be created with 7 touchdowns, yielding 21, and two field goals, yielding 10. 21+10= 31) 3. 22+10= 32 (32 can be created with 4 touchdowns, yielding 12, and 4 field goals, yielding 20. 20+12=22) 4. 23+10= 33 (33 can be created with 1 touchdown, yielding 3, and 6 field goals, yielding 30. 30+3=33) 5. 24+10= 34 (34 can be created with 8 touchdowns, yielding 24, and 2 field goals, yielding 10. 24+10= 34) 6. 25+10= 35 (35 can be created with 7 touchdowns, yielding 35) 7. 26+10= 36 (36 can be created with two touchdowns, yielding 6 points, and 6 field goals, yielding 30. 30+6=36) 8. 27+10= 37 (37 can be created with 9 touchdowns, yielding 27 points, and two field goals, yielding 10. 27+10=37) 9. 28+10= 38 (38 can be created with 1 touchdown, yielding 3, and 7 field goals, yielding 35. 35+3=38) 10. 29+10= 39 (39 can be created with 3 touchdowns, yielding 9 points, and 6 field goals, yielding 30. 30+9= 39. 11. 30+10= 40 (40 can be created with 5 touchdowns, yielding 15, and 5 field goals, yielding 25. 25+15=40 Every single number in this process just added 10, which represents two field goals. This process works because 10 is a possible combination, being two field goals.