++d +++d

//Ever wondered how many squares there are in a grid of squares?
//How about rectangles?
//Inspired by this image:  http://krexy.com/how-many-squares
//Found in many places online.

//Much thanks to LinqPad for providing such a nice dev environment.

//Basic Formula's - simple programming
Func<int,int> boxQuery = d =>
    (from x in Enumerable.Range(0,d)
    select (d-x) * (d-x)).Sum();

Func<int,int> rectQuery = d =>
    (from x in Enumerable.Range(0,d)
    from y in Enumerable.Range(0,d)
    select (d-x) * (d-y)).Sum();

(boxQuery(4) + boxQuery(2)*2).Dump();  //40 - the answer to the given image.
(rectQuery(4) + rectQuery(2)*2).Dump(); //118 - the same image, but including rectangles.

//Pure mathematical representations [thanks to WolframAlpha for providing basic formula]
Func<int,int> boxWolfPure = d => {d++; return ((2 * d)-1) *(d*(d-1))/6 ;};
Func<int,int> rectWolfPure = d =>  {d++; return ((d-1) * (d*d*d - d*d))/4;};

//Mixed math/programming - makes use of state [all d's after and inc ++d are actually (d+1)]
//Probably an abuse of code.
//Func<int,int> boxWolf = d => d*++d*(2*d-1)/6;
Func<int,int> boxWolf = d => d*(d +++d)*d/6;
Func<int,int> rectWolf = d => d*d*++d*d/4;

for (int i=0; i<100; i++) (boxQuery(i) + " " + boxWolf(i) + " " + rectQuery(i) + " " + rectWolf(i)).Dump();