Problem Solving

In the game of chess, the Knight can make any of the moves displayed in the diagram. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?

(A) 8
(B) 24
(C) 38
(D) 48
(E) 56

For Knight to be able to make all 8 moves.. it should be atleast 2 blocks away from each edge..
So the number of blocks where it can make all 8 moves are = (8-4)^2 = 16

Number of blocks from where all 8 moves are not possible are = 8^2 - 16 = 64-16 = 48

Option D

Knight can make all the moves displayed in the diagram only if it is in the squares left in white(In the diagram) = 4*4 = 16

Total number of squares = 8*8 = 64

So, the greatest number of spaces from which not all 8 moves are possible is 64-16 = 48

IMO D

Can someone please help me understand this question and the solution?

shankar.ashwin wrote:In the game of chess, the Knight can make any of the moves displayed in the diagram. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?

(A) 8
(B) 24
(C) 38
(D) 48
(E) 56

Since a chessboard is 8 by 8, the total number of positions on a chessboard = 8*8 = 64.
GOOD positions for the knight = 64 - BAD POSITIONS FOR THE KNIGHT.
Here, a BAD position is one from which the knight can be make all of the moves shown in the diagram.

In the diagram, the knight is positioned at the CENTER OF A 5 BY 5 SQUARE..
Implication:
To be able to make all of the moves shown in the diagram, the knight must be at the CENTER OF A 5 BY 5 SQUARE, as shown here:
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XXKXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX

Count how many 5 by 5 squares can be formed from the 64 positions on a chessboard.

If row 5 provides the bottom of the 5 by 5 square, we get the following options.
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XXXXXXXX
XXKXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX

XXXXXXXX
XXXXXXXX
XXXKXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX

XXXXXXXX
XXXXXXXX
XXXXKXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX

XXXXXXXX
XXXXXXXX
XXXXXKXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
Total number of options = 4.

If row 6 provides the bottom of the 5 by 5 square, there will be 4 more options.
If row 7 provides the bottom of the 5 by 5 square, there will be 4 more options.
If row 8 provides the bottom of the 5 by 5 square, there will be 4 more options.

Thus:
The total number of 5 by 5 squares that can be formed = 4+4+4+4 = 16, implying that there are 16 BAD positions from which the knight can make all of the moves shown in the diagram.
Since there are 16 bad positions, the total number of GOOD positions = 64-16 = 48.

The correct answer is D.

shankar.ashwin wrote:In the game of chess, the Knight can make any of the moves displayed in the diagram. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?

(A) 8
(B) 24
(C) 38
(D) 48
(E) 56

I guess it's pretty simple this way.

For a knight to have all 8 moves possible the square where it is located should be atleast 2 rows away from topmost row, 2 rows away from bottom-most row, 2 columns away from rightmost column and 2 columns away from left-most column.

Therefore all 16 blocks (as shown in figure within red boundaries) are possible blocks from where all 8 moves for a knight are possible.

Therefore remaining blocks from where all 8 moves are not possible = 64-18 = 48

Answer: Option D