A chessboard is an 8×8 array of identically sized squares. Each square has a particular designation, depending on its row and column. An L-shaped card, exactly the size of four squares on the chessboard, is laid on the chessboard as shown, covering exactly four squares. This L-shaped card can be moved around, rotated, and even picked up and turned over to give the mirror-image of an L. In how many different ways can this L-shaped card cover exactly four squares on the chessboard?
Options.
A) 256  B) 336  C) 424  D) 512  E) 672
Select 3 squares which are in a line together first. We will call them a block of 3 squares. Consider just the vertical arrangement for now (for horizontal, we will multiply everything by 2 at the end).   Consider a block of 3 squares lying vertically on the left edge (the first column of the chess board). You can place another square on the right to make an L at either extreme of the block. So for each such block, you can make an L in 2 ways. There will be 12 such blocks (6 on either edge).   You get 12*2 Ls.   Now consider the blocks of 3 squares lying vertically in columns 2 to 7. You can make an L by placing a square on left or right at either end. So for each block of 3 squares, you can make 4 Ls. There are 6 such blocks in each of the 6 columns   You get 6*6*4 Ls.   Total you get   12*2 + 6*6*4= 
<gadget id="calculator">  12*2 + 6*6*4</gadget>
<output>168</output>
168 Ls.   Now you just multiply it by 2 to account for the Ls lying horizontally too. Since it is a square, the number of Ls found vertically will be the same as the number of Ls found horizontally.   Total= 168*2= 
<gadget id="calculator">168*2</gadget>
<output>336</output>
336   Answer (336)
<result>B</result>
A chessboard is an 8×8 array of identically sized squares. Each square has a particular designation, depending on its row and column. An L-shaped card, exactly the size of four squares on the chessboard, is laid on the chessboard as shown, covering exactly four squares. This L-shaped card can be moved around, rotated, and even picked up and turned over to give the mirror-image of an L. In how many different ways can this L-shaped card cover exactly four squares on the chessboard?