What function when given the inputs $x,y$ returns the given $z$? What function when given the inputs $x, y$ returns the given $z$? 


*

*When $x = 2, y = 10$, $z = 1$

*When $x = 6, y = 10$, $z = 2$

*When $x = 50, y = 70$, $z = 5$

*When $x = 16, y = 17$, $z = 1$

*When $x = 1, y = 3$, $z = 0$

*When $x = 20, y = 30$, $z = 3$

*When $x = 100, y = 140$, $z = 9$


I'm trying to find if there is a relation between these examples.
 A: The problem is not well defined. There need to be more constraints. I can choose for example $f(x,y)=a_0x^6+a_1x^5+a_2x^4+a_3x^3+a_4x^2+a_5x+a_6$. Note that I did not have to use $y$ at all. If I plug in your 7 conditions, I get the $a_i$ coefficients. You can replace some of the terms with some expressions depending on $y$ to some powers, or a mixture of $x$ and $y$. You cannot have an expression that is independent of x, since the first two conditions have the same $y$ and different $z$. Or I can define a trivial function $f(x,y)=0$ except the following 7 cases: f(2,10)=1, f(6,10)=2,...
A: All seven of the $x$ values are different. We can disregard the $y$ values. But 
    For $1\le n\le7$ let $x_n,\,y_n,\,z_n$ denote the given values and define
    \begin{equation}
 g_n(x,y)=z_n\prod_{k\ne n}\frac{(x-x_k)}{(x_n-x_k)}
 \end{equation}
    Then for $m\ne n$
    \begin{equation}
 g_n(x_n,y_n)=z_n
 \end{equation}
    \begin{equation}
 g_n(x_m,y_m)=0
 \end{equation}
    Then define
    \begin{equation}
 f(x,y)=\sum_{n=1}^{7}g_n(x,y)
 \end{equation}
Then for $1\le n\le7$,  $f(x_n,y_n)=z_n$.
