Want to try prove the above using induction. I have worked out the largest impossible score is 11, thus meaning any score 12 or larger should be possible.
Where I'm struggling is to actually come up with some sort of expression to prove. It is my instinct to say that any score above 12 should be divisible by a linear combination of 7x + 3y for some x and y that are natural numbers.
Can anyone help point me in the right direction to begining my induction proof?
This is the Frobenius number problem, and for two values $x$ and $y$ with $\gcd(x,y)=1$ the desired number is $x y -x-y$, which is $11$ when $x=3$ and $y=7$.
We wish to prove that every integer greater than $11$ can be expressed in the form $7x + 3y$, where $x$ and $y$ are positive integers. Let's look at the first few cases. \begin{align*} 12 & = 3 + 3 + 3 + 3\\ 13 & = 7 + 3 + 3\\ 14 & = 7 + 7\\ 15 & = 3 + 3 + 3 + 3 + 3 \end{align*} Notice that if $n \geq 12$ has at least two summands equal to $3$, we can obtain $n + 1$ by replacing two $3$'s with a $7$. If the expression for $n \geq 12$ contains fewer than two $3$'s, then we must have at least two $7$'s, so we can obtain $n + 1$ by replacing two $7$'s with five $3$'s.
Alternatively, observe that one we establish that $12, 13, 14$ can be expressed in the form $7x + 3y$ for positive integers $x$ and $y$, we can establish that $n \geq 15$ can be expressed in that form by observing that if $n - 3 = 7x + 3y$, then $n = 7x + 3(y + 1)$.
