## Chess Anyone

How many rectangles of any size are on an $8\times 8$ checker board?
Source: mathcontest.olemiss.edu 2/8/2010

SOLUTION
A smaller chessboard
$4\times 4$

How many $1\times 2$ rectangles in a $4\times 4$ chessboard?
First, we draw a $1\times 2$ rectangle called the base rectangle at the upper left corner of the chessboard. We notice that there are $2$ empty units under it. We move the base rectangle down one unit at a time to form two additional $1\times 2$ rectangles for a total of three. The pattern is $1+\left (4-2\right )=3$.
Likewise there are $3$ empty units to the right of the base rectangle. We move it to the right one unit at a time to form 3 additional $1\times 2$ rectangles for a total of four. The pattern is $1+\left (4-1\right )=4$.
In general, let $x\times y$ be a rectangle in a $n\times n$ chessboard, where $x$ denotes the horizontal dimension and $y$ denotes the vertical dimension. The number of possible rectangles is
$\left [1+\left (n-x\right )\right ]\left [1+\left (n-y\right )\right ]$
Examples
Number of possible $1\times 2$ rectangles in a $4\times 4$ chessboard; $x=1;y=2;n=4$
$\left [1+\left (4-1\right )\right ]\left [1+\left (4-2\right )\right ]=4\times 3=12$
Number of possible $2\times 1$ rectangles in a $4\times 4$ chessboard; $x=2;y=1;n=4$
$\left [1+\left (4-2\right )\right ]\left [1+\left (4-1\right )\right ]=3\times 4=12$
Notice that there are as many $1\times 2$ rectangles as $2\times 1$ rectangles.
A bigger chessboard $8\times 8$
Let’s use the general formula to calculate all possible rectangles of any size in an $8\times 8$ chessboard as follows

 1 2 3 4 5 6 7 8 Number of rectangles 1 64 56 48 40 32 24 16 8 288 2 56 49 42 35 28 21 14 7 252 3 48 42 36 30 24 18 12 6 216 4 40 35 30 25 20 15 10 5 180 5 32 28 24 20 16 12 8 4 144 6 24 21 18 15 12 9 6 3 108 7 16 14 12 10 8 6 4 2 72 8 8 7 6 5 4 3 2 1 36

The rows represent the horizontal dimension $x$ and the columns represent the vertical dimension $y$.
Examples
Number of $1\times 1$ rectangles
$\left [1+\left (8-1\right )\right ]\left [1+\left (8-1\right )\right ]=8\times 8=64$
Number of $1\times 2$ rectangles
$\left [1+\left (8-1\right )\right ]\left [1+\left (8-2\right )\right ]=8\times 7=56$
Total number of rectangles of any size in an $8\times 8$ chessboard
$288+252+216+180+144+108+72+36=1296$

Answer: $1296$