Efficient way for finding unknowns with less equations than unknowns

Question

I'm trying to solve a problem where I have more unknowns than equations. The question is "how many ways are there to achieve 28 points in basketball with 17 throws, given that each throw can yield one point, two points or three points".

I found a few solutions by just manually trying stuff out but I figured I could maybe represent this with some equations. I have formed this into some equations like this (using $t$ for 3-point, $d$ for 2-point and $s$ for 1-point throws):

• $3t + 2d + 1s = 28$
• $t + d + s = 17$

But there is of course more than one solution for this (so far I found $[t = 0, d = 11, s = 6]$, $[t = 1, d = 9, s = 7]$, $[t = 2, d = 7, s = 8]$, $[t = 3, d = 5, s = 9]$) and the equations really don't seem to help a lot when you have more unknowns than equations. So I wonder what would be an efficient way to approach this without having to try out all combinations. Also how can I be sure I found all solutions and did not miss one? I'm not so much interested in the actual solution but rather in finding the way to get there.

Thank you very much for your help!

Answer 1

You can use Gaussian elimination and arrive at

$$\left( \begin{array}{cccc} 1 & 0 & -1 & -6 \\ 0 & 1 & 2 & 23 \\ \end{array} \right)$$

This means you have a free variable "s" that can be anything.

We have $$t = -6 + s \\ d = 23 -2s$$

Now you just need to select values that meet your criteria.

We need to $t \ge 0$ which makes $s \ge 6$ and $d \ge 0$, so $s \le 11$.

This means you have six values because $6 \le s \le 11$.

Answer 2

Because you are working over the nonnegative whole numbers, you will have a finite number of solutions.

Start with $s = 0$ and see if you can solve the system for $t$ and $d$ (if a solution exists, there will only be one; and check that both $t,d \geq 0$).

Repeat for $s = 1,2...$ What is the largest value of $s$ you need to test?

I can see six solutions graphically.