With_every_step wrote:
There is only one square on a chess-board right now, and it's you.
damn
How Many Squares on a Chessboard?
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May 7, 2017
0
#43
macer75 wrote:
The_Ghostess_Lola wrote:
The real question is:
How many rectangles on a chessboard ?
No, the real question is:
How many triangles are there on a chess board?
what say do you have in this, measly 1800?
May 8, 2017
0
#45
ghost_of_pushwood wrote:
The_Ghostess_Lola wrote:
The real question is:
How many rectangles on a chessboard ?
8!x8!=1296
I believe you, but I can't proof visualize it. Can u help me ?....Thanx in advance !
204
Sum of r^2 of n=8, 204
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.
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.
[COMMENT DELETED]
May 9, 2017
0
#50
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.
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.
ez 204
94
May 10, 2017
0
#54
8! x 8! doesn't = 1296. (8 + 7 + .... + 1) x (8 + 7 + .... + 1) = 1296. Making it a summation, not a factorial.
8! x 8! = 1,625,702,400.
May 10, 2017
0
#55
Someone on the Internet says that:
A square is a special kind of rectangle, it is one where all the sides have the same length. Thus every square is a rectangle because it is a quadrilateral with all four angles right angles. However not every rectangle is a square, to be a square its sides must have the same length.
So. After we figure out how many rectangles are on a chessboard, then we can add the # of squares (204 ?).
There are :
1 8x8 square, 4 7x7 squares, 9 6x6 squares and so on (16 5x5, 25 4x4, 36 3x3, 49, 2x2, 64 1x1)
adding, 1,4,9...64 gives : 204.
It's a square pyramid
May 10, 2017
0
#58
The_Ghostess_Lola wrote:
Someone on the Internet says that:
A square is a special kind of rectangle, it is one where all the sides have the same length. Thus every square is a rectangle because it is a quadrilateral with all four angles right angles. However not every rectangle is a square, to be a square its sides must have the same length.
So. After we figure out how many rectangles are on a chessboard, then we can add the # of squares (204 ?).
no, because the squares were already counted...for example in the "visual" explanation, for the height of 1, 8 1x1 squares were counted in first row and mult by 8. so thats the 64 1x1 squares. etc etc
May 10, 2017
0
#59
May 10, 2017
0
#60
The_Ghostess_Lola wrote:
Someone on the Internet says that:
A square is a special kind of rectangle, it is one where all the sides have the same length. Thus every square is a rectangle because it is a quadrilateral with all four angles right angles. However not every rectangle is a square, to be a square its sides must have the same length.
So. After we figure out how many rectangles are on a chessboard, then we can add the # of squares (204 ?).
No, because squares are rectangles. Read the definition.
It's like saying count all the animals and count all the dogs. The dogs are animals.
If it was "count all the squares, and count all the non-squares" like "count all the dogs and count all the non-dogs" then you would add them.
It depends on what version of "Chess" you are playing.