In the game of chess, the Knight can make any of the moves displayed

We can solve it by just counting the boxes..
Anyway the logic behind counting and calculation is given below..

From the picture it is very clear that the Knight can take all the 8 moves only if it is not in the last 2 rows or columns of the board.
So lets calculate the number of spaces where all the 8 moves are not possible.

Last row an column : (8+8+6+6)= 28
Penultimate row and column : (6+6+4+4) = 20

Total = 28+20 =48

Answer D.

Chess board has '64' squares.

To have ALL 8 moves possible, from the reference point of knight's square, there should be movement of 2 steps feasible in each direction - meaning 2 places above, 2 places below, 2 places right and 2 places left. If these conditions are met, then all 8 moves will be possible.

If we look at the outermost 8 by 8 matrix (leftmost column, rightmost column, top row and bottom row) - we can easily decipher that many of those 8 moves wont be possible from any of those squares.

Now lets look at the 6 by 6 matrix (second column from left, second column from right, second row from top and second row from bottom), here also we can decipher that all 8 moves wont be possible from any of these squares.

Now lets similarly move to 4 by 4 matrix. Here all 8 moves will be possible - because there are 2 places to left, right, above and below available. Inside these also, from all these squares, 8 moves will easily be possible.
So all 8 moves are possible from these interior '16' squares.

Our required answer = 64 - 16 = 48. Hence D

Interesting problem, especially because I love to play chess. There are programs called "chess engines", that can help to find best move, for example Stockfish.

If the Knight is located at a space (i.e., square) that is either on the border of the chessboard or next to a square that is on the border of the chessboard, then not all 8 moves by the Knight are possible. The number of spaces that are on the border of the chessboard is 8 x 4 - 4 = 28, and the number of squares that are next to a square that is on the border of the chessboard is 6 x 4 - 4 = 20. Therefore, there are 28 + 20 = 48 such spaces from which not all 8 moves are possible.

Answer: D

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