How many squares on a chessboard?

Somebody spoke, and I went into a dream...

(How many rectangles are there on a chess board ?).....Hmmm....

1296.

No, sorry, 1297, there's a square border.

:)

wow 161 posts... this forums seems like a 2 or 3 post reply... Me and a couple of others answered this very early on... 

there are 204 squares on the chess board 

the math is right here

n x (n + 1) x (2n + 1)
______________________
6

this should do it.

as for rectangles there are 1296 and the math is here

Count the rectangles by rows. In row 1 there are n 1 by 1s , n-1 1 by 2s, ... two 1 by n-1s and one 1 by n. Row Total = n + (n-1) + (n-2) + ... + 3 + 2 + 1 = n(n+1)/2 rectangles. But of course there are n rows, giving n (row sum). Now count all rectangles of height 2. Start in the bottom row. There are n 2 by 1s and n-1 2 by 2s and ... and one 2 by n. Row Total = n + (n-1) + (n-2) + ... + 3 + 2 + 1 = n (n+1)/2 The row total is the same, as it will be for all the rectangles of height 3, 4, ... n because they all share the same bases at the bottom of the board. However, there are only n-1 rows of height 2 and n-2 rows of height 3 etc. Thus, number 1 by any totals: n [n(n+1)/2] number 2 by any totals: (n-1) [n(n+1)/2] number 3 by any totals: (n-2) [n(n+1)/2] ... number n by any totals: 1 [n(n+1)/2] So the total number of rectangles is [n(n+1)/2](n + n-1) + ... + 3+ 2+1) or Total = [n(n+1)/2]^2 

then of course you just put 8 into the formula to get the answer...

another way to do it (does not require algebra) you just add the cubes of 1 through 8... I am not really sure why this works but if you have figured 

this way out then good for you =)

I know... we also took care of that on useful comment 4...

Wow !....good one on #162 ! Thx 4 the answer....and since noone's disputing u then u must be right !

lol of course im right =) I major in math =)

[COMMENT DELETED] but edited afterwards. lol

lol my post really put an end to this thread huh....

[COMMENT DELETED]and edited after again

have a look at this article about chess board perimeter - theres a table with how many rectangles ( the answer is 1296 ) , squares ( here it would be 204) there are on the chess board, as well as what is and how many squares can you fit into the perimeter of chess board ( this seems to be quite common question in various types of quizzes - let me know what you think

204