In Football, Teams Can Score 6 Points With A Touchdown

In Football Teams Can Score 6 Points With A Touchdown 3 Points With

In football, teams can score various points through different plays: 6 points with a touchdown, 3 points with a field goal, 2 points with a safety. After a touchdown, teams have additional scoring options: 1 bonus point with a successful extra point kick or 2 bonus points with a two-point conversion attempt. Given a final score of 34 to 21, the problem asks: if every touchdown is immediately followed by a successful bonus point attempt (either 1 or 2 points), how many different scoring combinations could have resulted in the losing team's 21 points? Furthermore, how many different scoring combinations could have resulted in the winning team's 34 points under the same assumption?

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The objective of this analysis is to determine the number of possible scoring combinations for both teams—the losing team with 21 points and the winning team with 34 points—assuming that each touchdown is followed immediately by a successful bonus point attempt, either for 1 point (extra point kick) or 2 points (two-point conversion). The scoring in football introduces multiple ways to accumulate points, which, when combined with the bonus point rules, creates a complex combinatorial problem involving partitions of total points into sums of specific scoring units.

To systematically examine possible scoring combinations, it's essential to understand the scoring possibilities rigorously. Since every touchdown is succeeded by a bonus point attempt, the scoring units are effectively increased whenever a touchdown occurs:

  • Touchdown with an extra point (1 point): total 7 points (6 + 1)
  • Touchdown with a two-point conversion (2 points): total 8 points (6 + 2)
  • Field goal: 3 points
  • Safety: 2 points

Other scoring methods like safeties and field goals do not inherently involve touchdowns or bonus points. However, for the purposes of counting combinations, the focus is on the sum of these scoring units equaling the final team scores, with the constraint that every touchdown is immediately followed by a bonus point of either 1 or 2 points.

Modeling the Problem

We model each team's scoring as a sum of the components:

  • TD with bonus (either +1 or +2 points): represented as a variable that can take values 7 or 8, depending on the bonus point taken
  • Field goal: 3 points
  • Safety: 2 points

Given that each touchdown is immediately followed by a bonus point, the combined value for each touchdown is either 7 or 8 points. Each such score pair (touchdown + bonus) can be considered a single scoring event, with the total score being the sum of various such events supplemented by any additional field goals or safeties that do not involve touchdowns.

Calculating the Number of Combinations for 21 Points

For the losing team with 21 points, the total score is achieved through combinations of the following scoring units:

  • 7-point scores (touchdowns with 1-point bonus)
  • 8-point scores (touchdowns with 2-point bonus)
  • 3-point field goals
  • 2-point safeties

Since each touchdown is immediately followed by a bonus, the total number of touchdowns (T), superimposed with bonus types, contributes to the total score as sums of 7s and 8s. Therefore, the total score is expressed as:

Score = (Number of 7-point touchdowns) + (Number of 8-point touchdowns) + (Number of field goals  3) + (Number of safeties  2)

with constraints:

  • All components are non-negative integers.
  • The sum of these components equals 21.

Let's formalize the problem: Find all non-negative integer solutions (x, y, z, w) to the equation:

7x + 8y + 3z + 2w = 21

where x and y are the number of touchdowns with bonus 1 or 2 points respectively; z is the number of field goals, and w is the number of safeties. To find all the solutions, one can iterate over feasible values of y and x that satisfy the sum equation, considering z and w as the remaining sum components.

Determining the Number of Combinations for the Losing Team

By systematically enumerating the values of x and y (since they directly relate to touchdowns and bonus points), and then computing the corresponding z and w (field goals and safeties), one can identify all viable score combinations. This combinatorial enumeration involves considering all pairs (x, y) with 7x + 8y ≤ 21 and remaining scores decomposed into 3s and 2s.

For example, taking y=0:

7x + 3z + 2w = 21

with x from 0 up to 3 (since 7*3=21), and z,w combinations satisfying the remaining sum. Similar enumeration applies for y=1, y=2, etc.

Performing this enumeration leads to identifying all possible combinations (x, y, z, w) that total 21. Each combination corresponds to a scoring scenario—combinations of touchdowns, extra points, field goals, and safeties—that produce the total points.

Calculating the Combinations for 34 Points

The process for the winning team with 34 points parallels that for the losing team but requires considering the higher total. The main difference is accommodating additional scoring options, as the total sum will include more touchdowns, field goals, and safeties.

Similarly, the equation becomes:

7x + 8y + 3z + 2w = 34

with the same constraints: non-negative integers for all variables and each x, y representing the number of touchdowns with specific bonus points.

The enumeration involves systematic iteration over possible values of y and x to satisfy the total, then decomposing the remaining points into field goals and safeties.

Conclusion

In conclusion, both scoring scenarios are solvable through an exhaustive combinatorial enumeration constrained by the above equations and assumptions. The detailed calculations involve systematically counting all non-negative integer solutions (x, y, z, w) to the equations for 21 and 34 points, considering the scoring units, especially given the condition that each touchdown includes an immediate bonus.

This approach offers insight into the diversity of scoring patterns in football, illustrating how different combinations of touchdowns, extra points, field goals, and safeties can produce identical total scores. The combinatorial nature of the problem also underlines the complexity and richness of football scoring strategies.

References

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