Riddler Express: January 6, 2023

David Ding

January 6, 2023

Basketball Numbers

Explanation:

Let $x$ be the number of three pointers you need to score and $y$ be the number of two pointers. Please note that $x$ and $y$ must be non-negative integers. Then we have:

$$\begin{array}{rl}3x+2y& =N\end{array}$$

The above is an example of a linear Diophantine equation. The key to solving those equations when they are linear is to make a table and think critically about the conditions that satisfy those equations. For the one that we have, we note that $2y$ is always even. If $N$ is an even number like 60, then $3x$ must be even as well to make the sum even. Since 3 is odd, that means $x$ must be even, and so our table looks like the following for $N=60$:

Therefore, for $N=60$, there are 11 ways to score it with 3-pointers and 2-pointers with $x=0,2,4,6,8,10,12,14,16,18,20$.

The key here is to realize that we cannot score 1-pointers, and so if we have $N=61$, then $x=20$ is out of the question as we would have 60 points and no way to make an additional 1 pointer. Also, here since $N$ is odd, we need to have $x$ being odd as well. This means, by similar analysis for our linear Diophantine equation, that $x=1,3,5,7,9,11,13,15,17,19$ for only 10 ways, which is less than 11.

It is also clear that for any $N>61$, we would have at least 11 ways to score that many points. For example, for $N=62$, the cases are the same as that for $N=60$ with the number of 3-pointers. For $N=63$, since it is divisible by 3, we have an extra case with $x=21$ and $y=0$. Any integer greater than 63 would introduce more cases for $x$ and $y$.

Therefore, $N=61$ is the only answer!