# squares

Jay Peatea
Posers and Puzzles 01 Jun '03 12:19
1. 01 Jun '03 12:19
Here is an easy one I remember from school.

How many squares are there on a chess board ?
2. 01 Jun '03 15:56
Originally posted by Jay Peatea
Here is an easy one I remember from school.

How many squares are there on a chess board ?
8x8 + 7x7 + 6x6 + 5x5 + 4x4 + 3x3 + 2x2 + 1x1
I think.
No idea what it adds up to!
3. royalchicken
CHAOS GHOST!!!
01 Jun '03 16:39
Right on...of course there must be 64+49+36+25+16+9+4+1=204 squares . incidentally, when summing consecutive square number like this, there is a simple formula. Can anyone figure it out?
4. 01 Jun '03 18:04
Originally posted by Jay Peatea
Here is an easy one I remember from school.

How many squares are there on a chess board ?
After long thinking...8 times 8 = 64...that times 2 beacuse a chess bord has two sides = 128!

Olav
5. 01 Jun '03 18:14
6. 01 Jun '03 20:152 edits
Originally posted by LivingLegend
After long thinking...8 times 8 = 64...that times 2 beacuse a chess bord has two sides = 128!

Olav
Ah..I didn't understand the question...π There can also be squares of 2x2 and 3x3 etc.....don't have time to calculate thatπ
7. royalchicken
CHAOS GHOST!!!
01 Jun '03 22:38
We've all come up with 204.
8. 02 Jun '03 14:591 edit
Originally posted by royalchicken
Right on...of course there must be 64+49+36+25+16+9+4+1=204 squares . incidentally, when summing consecutive square number like this, there is a simple formula. Can anyone figure it out?
I don't have the time to give a proof to this, but I'm sure Acolyte will.

As you've figured out, the number of squares on a 8x8 chess board is:
1^2 + 2^2 + 3^2 + ... + 8^2 = 204

For a nxn chess board, it would be:
1^2 + 2^2 + ... + (n-1)^2 + n^2 = 1/6*n*(2n+1)*(n+1)

- Johan
9. royalchicken
CHAOS GHOST!!!
02 Jun '03 20:42
You're right, and we don't even need Acolyte to prove it; it is easily feasible by simple induction....Good job π!