Is there a pattern?
The rules of basketball were changed so that three pointers are worth five points and two pointers are worth three points. What is the largest number of points that a team cannot possibly obtain?
For example, a team can can obtain eight points (by scoring one five-pointer and one three-pointer), but not seven points.
The answer is 7.
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1 solution
First note that the smallest number of points a team can obtain in one go is three. One of the implications of this is that if a team can obtain n points, it can also obtain n + 3 points, by first scoring n and then scoring a three-pointer.
In the example, a team can obtain eight points by scoring a five-pointer and a three-pointer. This implies that a team can also obtain 11 points, by first scoring that five- and three- pointer, and then scoring a three-pointer afterwards.
Also note that a team can score nine points (three three-pointers) and ten points (two five-pointers). By having this "streak" of three consecutive obtainable points, we can say that every other number of points can be obtained.
After obtaining eight, nine, or ten points, a team can obtain eleven, twelve, or thirteen points. And after obtaining eleven/twelve/thirteen points, a team can obtain fourteen/fifteen/sixteen points, and so on!
Therefore, after seven points (which cannot be obtained), every number can be obtained.
Hence, 7 is the largest number of points that a team cannot obtain.