TITLE: Area of greatest integer function
QUESTION [6 upvotes]: Question:

Find the area enclosed by the function:
  $$\left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3$$
  where $\lfloor\cdot\rfloor$ represents the greatest-integer function.

I believe the easiest way to solve this problem would be to graph the function. However, I was unable to do so. Please note that this was a question on an examination and there was no graphing equipment available.

REPLY [3 votes]: We can separate it into cases :
$$\left\lfloor\frac{|3x+4y|}{5}\right\rfloor+\left\lfloor\frac{|4x-3y|}{5}\right\rfloor=3$$
$$\iff \left(\left\lfloor\frac{|3x+4y|}{5}\right\rfloor,\left\lfloor\frac{|4x-3y|}{5}\right\rfloor\right)=(0,3),(1,2),(2,1),(3,0)$$
Now note that 
$$\left\lfloor Z\right\rfloor=a\iff a\le Z\lt a+1$$
and that $\frac{|3X+4Y|}{5},\frac{|4X-3Y|}{5}$ represents the distance between a point $(X,Y)$ and the line $3x+4y=0, 4x-3y=0$ respectively. 
Since the two lines intersect perpendicularly, we have $16$ unit squares, and so the answer is $\color{red}{16}$.