chess boards

Fun question – How many squares are there on a chess boards ?

How many squares on a  Chess Board

The first, and most common response to this question is 64. There are 8 squares in each direction (8 in the rank and 8 in the file – to give them their correct notation) so 8 x 8 is 64.

The full answer however is a little more complex. There are in fact also squares of size 2×2, 3×3, 4×4, 5×5, 6×6, 7×7 and one large one of 8×8.

So how many of each are there ?

Well starting with the largest – 8×8 – of course there is only 1

With the 7×7 there 8 spaces of which the 7×7 square occupies 7, leaving a spare space in which it can be moved sideways. And of course there is an extra space in each direction. So there are 2 x 2 different possibilities for a 7×7 square to fit in the 8×8 grid.  So 4 x 7×7.

For the 6×6, this occupies 6 squares in each direction. But there are 8 available, so the square can be moved up or across in 2 more positions in each direction. So with a total of 3 options in each direction that’s 3×3=9 possible locations for our 6×6 square.

Using the same argument for the 5×5 square, there are 3 additional moves available in each direction, making a total of 4 each way, so then there are 4×4=16 variants of the 5×5 square

To complete the calculation there will be

5×5=25 positions for squares of size 4×4

6×6=36 positions for squares of size 3×3

7×7=49 positions for squares of size 2×2

8×8=64 positions for squares of size 1×1

So that’s a total of 1+(2×2)+(3×3)+(4×4)+(5×5)+(6×6)+(7×7)+(8×8)=1+4+9+16+25+36+49+64=204. Hence the answer is 2014 possible squares.

So that’s something to think about while you are waiting for your opponent to make his next move !

Going a little further

Of course you can then develop the argument to say how many rectangles are there on the board.  If you look at the board as a grid then there are 8 square in each direction but 9 edges or lines.

A rectangle needs to lines in each direction to form the shape , so the question is how many combinations of 2 are there in each direction – without using the same line on both sides.

C (n,r) is the general formula for this where ‘n’ would be the number of lines in total and r is the number of lines you want to select – and where you don’t want any repetition of the line chosen (otherwise there is no shape !).

This formula expands to n!/r!(n-r)!  where the exclamation mark is the factorial sign.

(if you are unfamiliar with the Factorial concept then very simply it is a series of numbers descending by 1 each time from the number shown that are then multiplied together. For example ‘factorial 4’ would be written as 4! and this is equal to 4x3x2x1=24.   ‘Factorial 6’ or 6! = 6x5x4x3x2x1=720.  )

So taking our example of the chess board, the formula C(n,r) equates to C(9,2) which can then be written as 9! / 2!(9-2)! = 9!/2!x7!   =  9x8x7x6x5x4x3x2x1/2x1x7x6x5x4x3x2x1 which cancels down to 72/2= 36 in each direction – so a combination possibility of 36 x 36 = 1296 rectangles in total on the board.

This of course also includes the squares (204 of them to be precise). So the number of rectangles that are not also squares is 1296 – 204 = 1092

A bit of useless information. But it’s a practical and relevant example for a chess board where you can put some of the mathematics to use that you never really understood the need for at school.  In fact by now you are probably wondering why you even interested in the answer in the first place !

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