Can You Change The Game?

Riddler Express

From Richard Jacobson comes a matter of bewildering basketball:

You and a friend are shooting some hoops at your local basketball court when she issues a challenge: She will name a number, which we’ll call N. Your goal is to score exactly N points in as many ways as possible using only 2-point and 3-point shots. The order of your shots does not matter.

For example, there are two ways you could score N = 8 points: four 2-pointers or two 3-pointers and one 2-pointer.

Your apparently sadistic friend chooses 60 for the value of N. You try to negotiate this number down, but to no avail. However, she says you are welcome to pick an even larger value of N. Does there exist an integer N greater than 60 such that there are fewer ways to score N points than there are ways to score 60 points?

Solution

I looked at some smaller values of N.

• There are zero ways you could score N = 1 points.
• There is one way you could score N = 2 points: one 2-pointer.
• There is one way you could score N = 3 points: one 3-pointer.
• There is one way you could score N = 4 points: two 2-pointers.
• There is one way you could score N = 5 points: one 3-pointer and one 2-pointer.
• There are two ways you could score N = 6 points: two 3-pointers or three 2-pointers.
• There is one way you could score N = 7 points: one 3-pointer and two 2-pointers.
• There are two ways you could score N = 8 points: four 2-pointers or two 3-pointers and one 2-pointer.
• $\cdots$

In order to achieve different ways of scoring an N value, the only substitution possible is two 3-pointers = three 2-pointers.

For N = 6k, for positive integer k, the number of ways is k+1.

There are 11 ways for N = 60.

Answer

61 = 54 + 7. There are 10 ways for N = 54. There is only one way for N = 7.

Thus, there are only 10 ways for N = 61.

Rohan Lewis

2023.01.07