I pick two whole numbers $x$ and $y$ between $1$ and $10$ inclusive (not necessarily distinct). My friend picks two numbers $x -4$ and $2y-1$. If the product of my friend's numbers is one greater than the product of my numbers, then what is the product of my numbers?
From the information given, we can construct the following equation: $xy + 1 = (x-4)(2y-1)$. This simplifies to $xy - x - 8y = -3$. We can then apply Simon's Favorite Factoring Trick and add $8$ to both sides to get $xy - x - 8y + 8 = 5$. This can be factored into $$(x-8)(y-1)=5$$Since $x\leq 10$, $x=9$ and $y=6$. Thus, the product of my two numbers is $9 \cdot 6 = \boxed{54}$.