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# A chessboard is an 8×8 array of identically sized squares. Each square

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Magoosh GMAT Instructor
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A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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24 Feb 2015, 16:11
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85% (hard)

Question Stats:

52% (02:17) correct 48% (02:12) wrong based on 206 sessions

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chessboard with knight-L tile.JPG [ 19.42 KiB | Viewed 3299 times ]

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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672

For a discussion of the techniques used in difficult counting problems, including the OE for this particular problem, see:
https://magoosh.com/gmat/2015/counting- ... -the-gmat/

Mike

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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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21 Mar 2015, 06:14
2
mikemcgarry wrote:
Attachment:
The attachment chessboard with knight-L tile.JPG is no longer available

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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672

For a discussion of the techniques used in difficult counting problems, including the OE for this particular problem, see:
https://magoosh.com/gmat/2015/counting- ... -the-gmat/

Mike

we have 8 ways of rotating/placing this L on the chess board and 7*6 combinations for each.

total = 7*6*8 = 336
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Untitled.png [ 81.19 KiB | Viewed 3069 times ]

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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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21 Jan 2016, 17:55
Can you please explain how you got 7*6?
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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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21 Jan 2016, 17:56
Can you please explain how you got 7*6?
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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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21 Jan 2016, 18:46
GoCoogs35 wrote:
Can you please explain how you got 7*6?

The way I got 7*6 was.

7 is the amount of way that the shape can fit going left to right. There are 8 squares but the first one cant be counted because it is 2 squares deep.

6 going from top to bottom. Counted using the first square and until the shape can be shifted down without going off the graph.

I think that is right. Someone correct if Im wrong.
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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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21 Jan 2016, 21:20
2
2
mikemcgarry wrote:
Attachment:
chessboard with knight-L tile.JPG

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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672

Here is another method of solving it:

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 = 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 = 336

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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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22 Jan 2016, 14:01
2
1
xLUCAJx wrote:
GoCoogs35 wrote:
Can you please explain how you got 7*6?

The way I got 7*6 was.

7 is the amount of way that the shape can fit going left to right. There are 8 squares but the first one cant be counted because it is 2 squares deep.

6 going from top to bottom. Counted using the first square and until the shape can be shifted down without going off the graph.

I think that is right. Someone correct if Im wrong.

Dear xLUCAJx,
I'm happy to respond. There are probably a dozen different ways to give a valid solution to this question. The wide-open scenario invites many different approaches. Your solution is 100% correct. Of course, the brilliant Karishma gave an elegant solution. Here's another way to conceptualize it, similar to Karishma's approach.

Consider the L in its current orientation, and start with it in the lower left corner.
Attachment:

L in corner.png [ 13.65 KiB | Viewed 2259 times ]

We can move this up so that the top square is in any of the five empty squares above it: those, plus this original, is 6 positions. Now, we can move any of these six to the right one space, then again, then again, until we have moved it to the sixth new space, when the right side of the L will come up flush against the rightmost boundary. That’s seven total columns, each with 6 positions, for a total of 42 while it is in this orientation.

Clearly, we can rotate by 90° clockwise, and we would have 42 new positions for that orientation. Then we can rotate by 90° clockwise again, and again. Four orientations, each with 42 positions, for 42*4 = 168 positions.

All of this is for the “forward L.” Now, if we pick up the card, and put it down flipped over, to get a “mirror image L,” this again will have 42 positions in each of 4 rotated orientations, for another 168 position.

The total is 2*168 = 336

This question comes from a blog with 11 other challenging counting problems:
https://magoosh.com/gmat/2015/counting- ... -the-gmat/
You can get more practice on problems of this sort here.

Does all this make sense?
Mike
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Re: A chessboard is an 8×8 array of identically sized squares. Each square  [#permalink]

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05 Oct 2018, 08:02
mikemcgarry wrote:
Attachment:
chessboard with knight-L tile.JPG

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?
(A) 256
(B) 336
(C) 424
(D) 512
(E) 672

Obs.: when I came here to post my solution, I found that it coincides with "thefibonacci" solution.
I will post it anyway, so that the readers have another "wording" (it may help)!

$$?\,\,\,:\,\,\,\# \,\,L - {\rm{shaped}}\,\,{\rm{positions}}$$

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.
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Re: A chessboard is an 8×8 array of identically sized squares. Each square &nbs [#permalink] 05 Oct 2018, 08:02
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