Consider the following riddle:
A ball game is played where the players can score with their feet (for $x$ points) or with their hands (for $y>x$ points). We know that $91$ scores are impossible, and among those is the score 48.
What are the values of $x$ and $y$ ?
Note: This is just for pleasure. I have a solution but am looking for the most elegant and concise alternatives.
Note that $x$ and $y$ must be co-prime, otherwise there would be an infinite number of impossible scores. For the co-prime case, the Wikipedia article on the Coin problem gives $(x-1)(y-1)/2$ as the number of non-representable integers (see Frobenius number for $n=2$). So $(x-1)(y-1)=182=2\cdot7\cdot13$.
Therefore $x-1$ must be a factor of $182$: $x-1=1,2,7,13,14,26,91,$ or $182$. So $(x,y)$ is one of: $$(2,183)$$ $$(3,92)$$ $$(8,27)$$ $$(14,15)$$ $$(15,14)$$ $$(27,8)$$ $$(92,3)$$ $$(183,2)$$
We are given that $48$ is an impossible score, which rules out $2, 3,$ and $8$. So we are left with $(x,y)=(14,15)$ or $(15,14)$.