### How Many Squares on a Chessboard?

Age Range: 7 - 11

Although this investigation seems quite simple, it requires a methodical approach if the correct answer is to be attained.

As the title suggests, the investigation involves children finding out how squares there are on a chessboard. You might think that there are only 64, but you would be wrong...

The diagram below shows that there are indeed 64 squares, you there are also some more...

Don't forget the one large square (shown in red here)...

And, also the 16 two-by-two squares shown below (although these aren't the only 2x2 squares!)...

There are many more different-sized squares on the chessboard.

The complete list of answers is shown below:

1, 8x8 square |

4, 7x7 squares |

9, 6x6 squares |

16, 5x5 squares |

25, 4x4 squares |

36, 3x3 squares |

49, 2x2 squares |

64, 1x1 squares |

Therefore, there are actually 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 squares on a chessboard! (in total 204).

A worksheet with a large chessboard which children can use to investigate this problem can be found below.

If the children manage to find all of them, ask them if they can see a pattern in the results (i.e. the square numbers in the table).

## Comments

## Connorsplanet

20th March 2011

the answers!?

## Doreen

19th November 2012

thanks.. it was clearly and simply explained :))

## bob

8th January 2013

i dont understand the explanation btw the answers 204 connorsplanet

## sgfsgsdg

1st March 2013

so basically what they are trying to show is that these squares can overlap (well actually they weren't showing this, but regardless). If they didn't then that's when the number 16 with the 2x2 squares would come into play, but if you look for all the squares whether they overlap or not there is 204P.S the 7x7 squares completely overlap each other yet within the whole chessboard there is still space for 4 of them

## lucy

4th May 2013

BRILLIANT! Thanks for helping me with my maths homework cause I needed the working out as well. LOL!

## Kaitlyn

30th October 2013

I do not know that how many squares on a chessboard because I don't play before but I want to learn to play this game. So please can you share the whole information about how to play this game with me here?

## Peter O'Kelly

18th March 2014

How many rectangles are there on a chess board?

## rukmagoud

4th November 2014

u can use the formula 1square+2 square+3 square+................+8 square as n(n+1) (2n+1)/6 to get the answer

## Sparsh Gupta

13th June 2015

1296

## Dory

17th July 2016

Great! Thanks for solving my college homework! Well explained

## Someone answer ...

17th August 2016

what about for a checkerboard that isn't 8x8? how do you figure it out for any other number; nxn checkerboard??

## Jeff

28th August 2016

What is the rule for finding how many squares are in any size chessboard?

## Sangita, teacher

4th December 2016

This is a great way for students to develop their mind and multiplication skills. I taught my students squared numbers, and then used this to test their knowledge!

Although I would recommend giving hints as it stresses the mind...

## Sangita, teacher

15th December 2016

93% worked out the answer of a grade 5 class.

## Kenedy Byarugaba

7th January 2017

You are absolutely perfect

## pavan

12th January 2018

You explained it in a very simple manner, Thanks a lot for your explanation

## Doggert

15th March 2018

Thank you for the answer. 204 by the way is not a square number.

## MattBang

16th August 2018

s = width of the square

xn = number of square/s that can exist horizontally

yn = number of square/s that

if s = 8 (the biggest square)

then xn = 1, yn = 1

total # of 8-length square = 1*1 = 1

if s = 7 (the 2nd biggest square)

then xn =2, yn = 2

total # of 7-length square = 2*2 = 4

if s = 6 (the 3rd biggest square)

then xn =3, yn = 3

total # of 6-length square = 3*3 = 9

and so on...

ergo 1*1 + 2*2 + 3*3 + 4*4 + 5*5 + 6*6 + 7*7 + 8*8 = 204

or 1²+2²+3²+4²+5²+6²+7²+8²

## Darren Bolton

4th September 2018

Hi, in answer to the question about deriving the number of squares for any size of square checkerboard... I have added this as a great 'challenge question' into my sequences lesson on cubic sequences. If you start with the single square in the top left corner, a 1x1 square (n=1) the number of possible squares is, obviously, 1. For a 2x2 (n=2) the number of squares is 5. Working this up to n=5, gives a cubic sequence of 1,5,14,30,55 which equates to an nth term of (2n^3 +3n^2 + n) / 6. If you try n=8 you will see that it gives the correct answer of 204. The benefit of generating an nth term expression is that it allows for calculations of the number of squares for checkerboards larger than 8x8.

## Susheela aravindan

17th October 2018

Excellent