How Many Squares on a Chessboard? - Chess Forums - Page 7

iballisticsquid123 · 2017-05-05T16:53:43-07:00

This is a trick question. Lets see if any of you can figure it out.

MathWizKidA

Apr 15, 2018

0

#121

ghost_of_pushwood wrote:

The_Ghostess_Lola wrote:

The real question is:

How many rectangles on a chessboard ?

8!x8!=1296

FYI, 8!=40320, and 40320^2=1625702400.

That seems like it's WAY TOO MANY!!!

MathWizKidA

Apr 15, 2018

0

#122

Enderman1323 wrote:

incorrectname wrote:

Enderman1323 wrote:

incorrectname wrote:

208 some thers a square around the square, then both sides

That's a rectangle

That is unless your chess board is a perfect cube

by sides i mean the top and bottom

Ok, who's to say I don't have a circular chess set?

Who said you did?

HalmaMarci

Apr 15, 2018

0

#123

😉

HorribleTomato

Apr 15, 2018

0

#124

204

MathWizKidA

Apr 15, 2018

0

#125

HorribleTomato wrote:

204

Hi! Why won't you let me do bulldog chess or something? I just want to...

Enderman1323

Apr 15, 2018

0

#126

MathWizKidA wrote:

Enderman1323 wrote:

incorrectname wrote:

Enderman1323 wrote:

incorrectname wrote:

208 some thers a square around the square, then both sides

That's a rectangle

That is unless your chess board is a perfect cube

by sides i mean the top and bottom

Ok, who's to say I don't have a circular chess set?

Who said you did?

You can't assume that the lining of someone's chess board is a perfect square. The burden of proof is on you to prove that every single chess board lining is a perfect square. Good luck.

iballisticsquid123

Jun 25, 2018

0

#127

have you considered 209?

Enderman1323

Jun 28, 2018

0

#128

iballisticsquid123 wrote:

have you considered 209?

We have. It's wrong.

GM_chess_player

Jun 28, 2018

0

#129

204 SQUARES!

wilsonga0

Jun 28, 2018

0

#130

0. There’s no such thing as a perfect square except in the metaphorical sense.

incorrectname

Jun 28, 2018

0

#131

What makes you think that a chessboard wouldn't be metaphorical? How about being used like a pawn on a chessboard?

incorrectname

Jun 28, 2018

0

#132

oops that was a simile

iballisticsquid123

Dec 29, 2018

0

#133

heres the explanation you've been waiting for. There are C9,2 or 36 ways to choose two vertical lines and the same for horizontal lines. multiply those together to get the legit answer of 1296 rectangles.

E_Shan

Mar 20, 2019

0

#134

Easy explanation.

8x8+7x7+6x6+5x5+4x4+3x3+2x2+1x1= 204

AlCzervik

Mar 23, 2019

0

#135

a whole lot more than 204!

iballisticsquid123

Jun 24, 2019

0

#136

The_Chin_Of_Quinn wrote:

Fireline11 wrote:

I thought 8!=8*7*6*5*4*3*2*1. Am i confused?

I do agree that there are 1296 recta gles on an 8x8 chessboard. I think the easiest way to visualize is bij first starting in the lowest row and fit rectangles in this row with a height of 1. Of course there are 8 1x1 rectangles, 7 1x2 rectangles... etc. So in total 1+2+3+4+5+6+7+8=36 rectangles. You have 8 rows so that already gives 8*36 rectangles.

Now what about height 2. You need to take the lowest two rows and fit rectangles in them. You can also do this in 36 ways. (You could visualise this in the same manner as before) .You can move each rectangle with height two up 6 times so in total that gives us 7*36 triangles with height 2.

You see where this is going, right? In total there are (1+2+3+4+5+6+7+8)*36=36*36=(40-4)(40-4)=1600-320+16=1296 rectangles :) if you have a chessboard with dimensions nxn, you will have
(1/4)*n^4 +(1/2)*n^3+(1/4)n^2 rectangles.

Nice, I like how you explain to visualize it.

It's interesting to me that 1296 can also be expressed as the sum of cubes from 1 to 8.

There is actually a formula from the sum of cubes from 1 to n. it goes, (1+2+3...+n)^2. try it!

MARattigan

Jun 25, 2019

0

#137

Oh dear. I think our readers spend too much time playing chess.

There are 9-r consecutive intervals of r squares both horizontally and vertically, so (9-r)² squares of side r. So the total number of squares is 1²+2²+...+8².

But 1²+2²+...+n² = ⅙n(n+1)(2n+1)

I'll leave you to work it out.

Edit: Sorry -only read the first page before posting.

MARattigan

Jun 25, 2019

0

#138

The_Ghostess_Lola wrote:

The real question is:

How many rectangles on a chessboard ?

(8(8+1)/2)² - again, leave you to work it out.

Edit: Sorry, too slow again.

MARattigan

Jun 25, 2019

0

#139

The_Ghostess_Lola wrote:

The real question is:

How many rectangles on a chessboard ?

The real real question is how many simple polygons with boundaries composed of sides of the small squares are there?

TimothyScottPuente

Jun 25, 2019

0

#140

A 64-square factorial puts the number in excess of billions.

Ciao,

Timothy Scott Puente