Football Brain Teaser!?

Question

The Chicago Bears score 18 points in a football game. In how many different ways can the Bears score these points? Points are scored as follows: A safety is 2 points, a field goal is 3 points and a touchdown is 6 points. After a touchdown is scored, the team may get 1 extra point for a kick, 2 extra points for a 2 point conversion or 0 points if they fail either of the above. If possible, please show your work and enjoy!

*Hint: Combine the conversion points to make the problem easier.

Answer 1

First, look at all the point combinations of TDs possible:
3 TD (0XP) = 18
2 TD (2XP) = 16
1 TD (1XP) + 1 TD (2XP) = 15
2 TD (1XP) = 14
1 TD (0XP) + 1 TD (2XP) = 14
1 TD (0XP) + 1 TD (1XP) = 13
2 TD (0XP) = 12
1 TD (2XP) = 8
1 TD (1XP) = 7
1 TD (0XP) = 6

Next, look at a matrix of safeties and FGs to see how many points each will produce (without going over 18). The difference between the total points of these combinations and 18 must equal one of the TD point combinations to get a successful combination totalling 18. Safety/FG combinations resulting in 1, 2, 3, 4, 5, 9, 10, 11 or 17 points left over will not be able to use a TD combo. 0 points left over will be valid combinations for safeties/FGs by themselves without needing TDs.

9 safeties, 0 FG, 18 points, 0 left over
8 safeties, 0 FG, 16 points, 2 left over
7 safeties, 1 FG, 17 points, 1 point left over
7 safeties, 0 FG, 14 points, 4 points left over
6 safeties, 2 FG, 18 points, 0 points left over
6 safeties, 1 FG, 15 points, 3 points left over
6 safeties, 0 FG, 12 points, 6 points left over, 1 TD (0XP)
5 safeties, 2 FG, 16 points, 2 points left over
5 safeties, 1 FG, 13 points, 5 points left over
5 safeties, 0 FG, 10 points, 8 points left over, 1 TD (2XP)
4 safeties, 3 FG, 17 points, 1 point left over
4 safeties, 2 FG, 14 points, 4 points left over
4 safeties, 1 FG, 11 points, 7 points left over, 1 TD (1XP)
4 safeties, 0 FG, 8 points, 10 points left over
3 safeties, 4 FG, 18 points, 0 points left over
3 safeties, 3 FG, 15 points, 3 points left over
3 safeties, 2 FG, 12 points, 6 points left over, 1 TD (0XP)
3 safeties, 1 FG, 9 points, 9 points left over
3 safeties, 0 FG, 6 points, 12 points left over, 2 TD (0XP)
2 safeties, 4 FG, 16 points, 2 points left over
2 safeties, 3 FG, 13 points, 5 points left over
2 safeties, 2 FG, 10 points, 8 points left over, 1 TD (2XP)
2 safeties, 1 FG, 7 points, 11 points left over
2 safeties, 0 FG, 4 points, 14 points left over, both 2 TD (1XP) and 1 TD (0XP) + 1 TD (2XP)
1 safety, 5 FG, 17 points, 1 point left over
1 safety, 4 FG, 14 poiints, 4 points left over
1 safety, 3 FG, 11 points, 7 points left over, 1TD (1XP)
1 safety, 2 FG, 8 points, 10 points left over
1 safety, 1 FG, 5 points, 13 points left over, 1 TD (0XP) + 1 TD (1XP)
1 safety, 0 FG, 2 points, 16 points left over, 2 TD (2XP)
0 safety, 6 FG, 18 points, 0 points left over
0 safety, 5 FG, 15 points, 3 points left over
0 safety, 4 FG, 12 points, 6 points left over, 1 TD (0XP)
0 safety, 3 FG, 9 points, 9 points left over
0 safety, 2 FG, 6 points, 12 points left over, 2 TD (0XP)
0 safety, 1 FG, 3 points, 15 points left over, 1 TD (1XP) + 1 TD (2XP)
0 safety, 0 FG, 0 points, 18 points left over, 3 TD (0XP)

To summarize,
9 safeties
6 safeties and 2 FG
6 safeties and 1 TD (0XP)
5 safeties, 1 TD (2XP)
4 safeties, 1 FG, 1 TD (XP)
3 safeties, 4 FG
3 safeties, 2 FG, 1 TD (0XP)
3 safeties, 2 TD (0XP)
2 safeties, 2 FG, 1 TD (2XP)
2 safeties, 2 TD (1XP)
2 safeties, 1 TD (0XP), 1 TD (2XP)
1 safety, 3 FG, 1 TD (1XP)
1 safety, 1 FG, 1 TD (0XP), 1 TD (1XP)
1 safety, 2 TD (2XP)
6 FG
4 FG, 1 TD (0XP)
2 FG, 2 TD (2XP)
1 FG, 1 TD (1XP), 1 TD (2XP)
3 TD (0XP)

19 combinations.

Had this been the Oakland Raiders, the solution would have been the null set-- don't think it's possible for them to score 18 points!

Answer 2

3 Touchdowns with all 3 extra points missed
6 field goals
2 touchdowns with both XPs missed, 2 field goals
1 TD w/ XP missed, 4 FGs
2 TDs, 2 2-point conversions, safety
2 TDs, 1 2-pt conversion, missed xp, 2 safeties
2 TDs, one with XP, one w/ 2pt conv, FG
TD w/ xp, safety, 3 FGs
9 safeties (yeah right!)
4 FGs and 3 safeties
2 FGs and 6 safeties
2 TDs with XPs missed, 3 safeties
1 TD w/ missed XP, 6 safeties
1 TD w/ XP, 3 FGs, safety
1 TD w/ XP, 1 FG, 4 safeties

This could go on forever!

Answer 3

To make 18 points: (use PPAT for 2 point conversion)
3 TD (miss all PAT)
2 TD, 2 FG (miss both PAT)
2 TD, 1 FG, 1 PPAT, 1 PAT
2TD, 3 SAF (miss all PAT)
2 TD, 2 SAF, 2 PAT
2TD, 2 PPAT, 1 SAF
1 TD, 2 FG, 1 PPAT, 2 SAF
1 TD, 4 FG
1 TD, 1 PAT, 3 FG, 1 SAF
1 TD, 1 FG, 4 SAF, 1 PAT
6 FG
4 FG, 3 SAF
2 FG, 6 SAF
9 SAF

... don't ~think~ I missed any choices ...

Answer 4

6 field goals
1 TD a missed extra point and 2 FG's

Answer 5

Bobby Charlton