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A chessboard is an 8X8 array of identically sized squares.

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A chessboard is an 8X8 array of identically sized squares.

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A chessboard is an 8X8 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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672

OA B

Source: Magoosh

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BTGmoderatorDC wrote:


A chessboard is an 8X8 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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672

Source: Magoosh
$$?\,\,\,:\,\,\,\# \,\,L - {\rm{shaped}}\,\,{\rm{positions}}$$
I was asked if there is a simple, error-avoiding, quick and "natural" approach (just that, LoL) to this very-nice problem.

There is! Although we will need to separate the problem in 8 configurations... all of them are trivial:

(The time I took to type the solution - drawing included - was approximately 15min.
But only 3min to find the solution to myself - ugly-hand-drawing included.)



Configuration 1: the "head" (guide-point in red) is up, the "tail" to the right.
We have 6 positions for the head in the first left viable column (I have shown the first and last in it)
We have 7 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities
(Numbers 6 and 7 are explained in the first drawing. The others below are analogous.)

Configuration 2: the "head" (guide-point in red) is up, the "tail" to the left.
We have 6 positions for the head in the first left viable column (I have shown the first and last in it)
We have 7 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

Configuration 3: the "head" (guide-point in red) is down, the "tail" to the right.
We have 6 positions for the head in the first left viable column (I have shown the first and last in it)
We have 7 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

Configuration 4: the "head" (guide-point in red) is down, the "tail" to the left.
We have 6 positions for the head in the first left viable column (I have shown the first and last in it)
We have 7 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

Configuration 5: the "head" (guide-point in red) is left, the "tail" to the right-down.
We have 7 positions for the head in the first left viable column (I have shown the first and last in it)
We have 6 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

Configuration 6: the "head" (guide-point in red) is right, the "tail" to the left-down.
We have 7 positions for the head in the first left viable column (I have shown the first and last in it)
We have 6 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

Configuration 7: the "head" (guide-point in red) is left, the "tail" to the right-up.
We have 7 positions for the head in the first left viable column (I have shown the first and last in it)
We have 6 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

Configuration 8: the "head" (guide-point in red) is right, the "tail" to the left-up.
We have 7 positions for the head in the first left viable column (I have shown the first and last in it)
We have 6 positions for the head in each row (I have shown the lower left and the lower right).
Multiplicative Principle: 6*7 = 42 possibilities

All cases above are exhaustive (i.e, cover all scenarios) and mutually exclusive (i.e., no double-countings), hence:
$$? = 8*42 = 336$$
This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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BTGmoderatorDC wrote:


A chessboard is an 8X8 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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672
Case 1:
Every 6-block rectangle with a base of 2 and a height of 3 yields 4 possible L-shapes:



Chessboard:

The base for the 6-block rectangle shown above can appear in rows 1 through 6, yielding 6 row options.
The bottom left corner for the 6-block rectangle shown above can appear in columns A through G, yielding 7 column options.
To combine the 6 row options with the 7 column options, we multiply:
6*7 = 42.
Since each of these 42 6-block rectangles will yield 4 possible L-shapes, we get:
Total options = 4*42 = 168.


Case 2:
Every 6-block rectangle with a base of 3 and a height of 2 yields 4 possible L-shapes:

Since the chessboard is square, Case 2 must yield the same number of options as Case 1:
Total options = 168.


Resulting total:
Case 1 + Case 2 = 168 + 168 = 336.

The correct answer is B.

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