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) 672Obs.: 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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Fabio Skilnik ::
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