# How many rectangles are there on an $8 \times 8$ checkerboard?

How many rectangles are there on an $8 \times 8$ checkerboard?

\begin{array}{|r|r|r|r|r|r|r|r|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{array}

Attempt

I just counted them up via casework: a $1 \times 1: 64$

$1 \times 2: 56$

$\vdots$

$1 \times 8: 8$ Then,

$2 \times 2: 49$

$2 \times 3: 42$

$\vdots$

The pattern continues like it looks like it should.

Thus, we can sum up all of these solutions as $8(1+\cdots+8)+7(1+\cdots+7)+\cdots+2(1+2)+1(1) = 750$, but the correct answer is $1296$. Where did I go wrong?

• Just as a comment, it's a little weird to me that the question which is interesting and well written has 3 upvotes while the answer (again well done) has 10 upvotes. Who are the people that upvoted the answer and not the question? Please consider giving the question some love too. – user223391 Dec 23 '15 at 3:54

$$\binom92^2=36^2=1296$$