Let your living room be the unit square. We denote the coordinates in the square by x and y. The dots are at locations $(x_i,y_j)$ (see the image here: https://i.stack.imgur.com/cPUTU.jpg). We denote the temperature in $(x_i ,y_j)$ by $u_{i,j}$. The temperature is the average of the temperatures in the four neighboring points: $$u_{i,j}=\frac{u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}}{4} \ (1).$$
Now assume that $u_{i,j}=u(x_i,y_j)$ with $u(x,y)=x^2-y^2$. Show that $(1)$ holds.
My attempt: The average of the temperatures in the four neighboring points is $$\frac{x_{i-1}^2-y_j^2+x_{i+1}^2-y_j^2+x_i^2-y_{j-1}^2+x_i^2-y_{j+1}^2}{4}.$$ We note that $\frac{x_{i-1}+x_{i+1}}{2}=x_i, \frac{y_{j-1}+y_{j+1}}{2}=y_j$. Then consider x coordinates first: $x_{i-1}^2+x_{i+1}^2+2x_i^2=(x_{i-1}+x_{i+1})^2-2x_{i-1}x_{i+1}+2x_i^2=4x_i^2-2x_{i-1}x_{i+1}+2x_i^2$. So we get $4x_i^2$, which is what we need, but I don't know how to deal with the rest...