In boffo Ball, a team can make two different kinds of scores: one that counts for 3 points and one that counts for 7 points. 1. In one game last year, the Mongooses defeated the Rattlesnakes by a score of 42 to 37. List all the ways that each team could have arrived at its final score. 2. Certain scores are impossible for one team to get - 4 points, for instance. List all the impossible scores you can. 3. Find the highest score that's impossible to get in Boffo Ball. Prove that it's the highest. For each of the three parts above: a. Explain how you went about solving the problem - what strategies you tried, what insights you had. b. Describe your solution as clearly as you can. c. Convince a reader of your paper that your solution is correct.
(1)Since the scores are in terms of either 3 or 7 or both …
The possible ways of scoring 42 for Mongooses are
(a) 3 * 14 (b) 7 * 6 (c) 3 * 7 + 7 * 3
The possible ways of scoring 37 for Rattlesnakes are
(a) 3 * 3 + 7 * 4 (b) 3 * 10 + 7 * 1
(2)
Yes, there are other ways such as
(a) 3 * 1 + 7 * 5.571 (b) 3 * 21 + 7(-3) for Mongooses and (a) 3 * 1 + 7 * 4.857 (b) 3 * 17 + 7 * -2 for Rattlesnakes, but these are not valid since the number of games cannot be decimals or negative numbers. Number of games should be a positive integer. Therefore, the only valid possibilities are those shown in (1) above.
The possible scores are multiples of positive multiples of 3, positive multiples of 7 and any combination of these two
Here is a partial list of possible scores: 3, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, …
(3)
The highest score that is impossible to get is 11
11 is the highest because all scores above 11, such as 12, 13 etc can be expressed in terms of 3 and 7. For example,
12 = 3 * 4, 13 = 3 * 2 + 7 * 1, 14 = 7 * 2, 15 = 3 * 5, 16 = 3 * 3 + 7, 17 = 3 * 1 + 7 * 2, etc
(a)
In solving the above problems, we can make equations 3x + 7y = 42 for Mongooses and 3x + 7y = 37 for Rattlesnakes, because the points earned can only be 3 and 7. x and y should take on only 0 or positive integer values (because number of games can’t be negative). We can then find x and y such that the combination of 3x and 7y makes 42 or 37.
(b) With the above constraints, the only (x, y) values possible to get 42 were found to be (14, 0), (0, 6) and (7, 3), and the only (x, y) values possible to get 37 were found to be (3, 4) and (10, 1)
(c) One can try out other possibilities whereby one can get 42 and 37 using only 3’s and 7’s. But it is not possible. Similarly, one can see that any score greater than 11 is possible as a combination of 3’s and 7’s. Thus the highest score that is impossible to get is 11.