In a football game, a touchdown with an extra point is worth 7 points and a field goal is worth 3 points. Suppose that in a game the only scoring done by teams are touchdowns with extra points and field goals.
A: Which of the scores 1 to 25 are impossible for a team to score?
B: List all possible ways for a team to score 40 points.
A: The scores 1, 2, 4, 5, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, and 25 are impossible for a team to score.
B: Possible ways for a team to score 40 points include:
- 6 touchdowns with extra points and 2 field goals
- 5 touchdowns with extra points and 3 field goals
- 4 touchdowns with extra points and 4 field goals
- 3 touchdowns with extra points and 5 field goals
- 2 touchdowns with extra points and 6 field goals
- 1 touchdown with extra point and 7 field goals
A: To determine which scores are impossible for a team to score, we can use the concept of the coin exchange problem. Let's consider the two possible scores, touchdown (7 points) and field goal (3 points).
We start by finding the greatest common divisor (GCD) of 7 and 3, which is 1. By using the Extended Euclidean Algorithm to calculate this GCD, we have: 7 * (-2) + 3 * 5 = 1.
Now, we can state that any score greater than or equal to (3 * 7) - 7 - 3 = 11 can be expressed by a combination of touchdowns and field goals, as the number of touchdowns or field goals can be negative (-2 touchdowns and 5 field goals).
Therefore, the scores 1, 2, 4, 5, 8, 10, 13, 16, 19, 22, 25 are impossible for a team to score.
B: To find all possible ways for a team to score 40 points, we need to consider the combinations of touchdowns and field goals.
Let's represent touchdowns as T and field goals as F.
Possible combinations can be:
1. TTTTTTTT
2. TTTTFFF
3. TTTFF
4. TTT
5. TFFF
6. FFFFFTTT
It's important to note that these are just a few examples, and there may be more combinations. But these represent a few possible ways for a team to score 40 points using only touchdowns and field goals.
To determine which scores are impossible for a team to achieve, we can use a concept called "Diophantine equations." In this case, we want to find the coefficients of touchdown (7) and field goal (3) that will allow us to form any number between 1 and 25.
To find the coefficients, we can use the "Chicken McNugget theorem" or the "Frobenius coin problem." This theorem states that for any two relatively prime positive integers, the largest number that cannot be expressed as their linear combination is their product minus their sum.
In our case, the touchdown value is 7 and the field goal value is 3. Their product is 21 (7 * 3), and their sum is 10 (7 + 3). Therefore, the largest number that cannot be achieved as a combination of touchdowns and field goals is 21 - 10 = 11.
Now, let's answer the questions:
A: Which scores between 1 and 25 are impossible for a team to score?
To find the impossible scores, we need to check if each number between 1 and 25 is greater than 11. The scores 12 to 25 are possible, but the scores 1 to 11 are impossible.
B: List all possible ways for a team to score 40 points.
To find all possible ways to score 40 points, we can use a combinatorial approach. We need to consider all combinations of touchdowns and field goals that sum up to 40.
Here are all the possible combinations:
- 5 touchdowns and 5 extra points
- 2 touchdowns and 10 field goals
Thus, a team can score 40 points by either scoring 5 touchdowns with 5 extra points, or by scoring 2 touchdowns with 10 field goals.