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# The number of rectangles that can be formed on a 8X8 chessbo

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Manager
Joined: 31 May 2010
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The number of rectangles that can be formed on a 8X8 chessbo  [#permalink]

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Updated on: 09 Jul 2013, 07:04
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Difficulty:

95% (hard)

Question Stats:

30% (02:29) correct 70% (02:13) wrong based on 36 sessions

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1. The number of rectangles that can be formed on a 8X8 chessboard is
A. 2194
B. 1284
C. 1196
D. 1296

D.

2. The number of squares on 8X8 chessboard is
A. 204
B. 220
C. 240
D. 210

A.

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Originally posted by saurabhgoel on 13 Sep 2010, 23:43.
Last edited by Bunuel on 09 Jul 2013, 07:04, edited 2 times in total.
Renamed the topic, edited the question and added the OAs.
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Updated on: 14 Sep 2010, 05:25
For qtn2: # SQUARES
answer = 1^2+2^2+3^2 . . . . 8^2 = 204 ...aplly the formula n(n+1)(2n+1)/6 for the SUM of squares of the first n #s ...n = 8 in this case).

Explanation:

To make a square from a 8by8 grid, we need to select equal # of grids from row and column.

8 adjucent grids from row and 8 adjucent grids from columns = 1*1 = 1^2
7 adjucent grids from row and 7 adjucent grids from columns = 2*2 = 2^2
.
.
.
.
1 adjucent grid from row and 1 adjucent grid from columns = 8*8 = 8^2

total = 1^2+2^2+3^2 . . . . 8^2

Originally posted by muralimba on 14 Sep 2010, 05:13.
Last edited by muralimba on 14 Sep 2010, 05:25, edited 1 time in total.
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14 Sep 2010, 05:24
No qtn 2 -rectangles = (asuumed SUQARES(n by n) and RECTANGLES (m by n))

# of ways to select 1 grid from a cloumn = 8
now i can select 1/2/3/4/5/6/7/8 adjucent grids from the row to form a suqare/rectangle

hence # of ways = 8 (1+2+3+.....+8)
so onnnn

total = 8 (1+2+3+.....+8) + 7 (1+2+3+.....+8) + ........ + 1 (1+2+3+.....+8)

= (1+2+3+.....+8) * (1+2+3+.....+8) = (1+2+3+.....+8)^2 = 36^2 = 1296

Hence answer = 1296 ((asuumed SUQARES(n by n) and RECTANGLES (m by n))

if asked for m by n (m not=n) rectangles only then answer = 1296-204 (# of squares as calculated in the above post)

Hope it is clear
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14 Sep 2010, 07:14
1
2
Another way:

1. The number of rectangles that can be formed on a 8X8 chessboard is
a) 2194
b)1284
c) 1196
d) 1296

8 rows and 8 columns are separated by 9 vertical and 9 horizontal lines. Any 2 vertical line and any 2 horizontal line will make a rectangle, so total # of rectangles possible is $$C^2_9*C^2_9=36*36=1296$$.

2. The number of squares on 8X8 chessboard is
a) 204
b) 220
c) 240
d) 210

# of squares with are of 1*1=1 is 8*8=64;
# of squares with are of 2*2=4 is 7*7=49;
# of squares with are of 3*3=9 is 6*6=36;
...
# of squares with are of 8*8=64 is 1*1=1;

Total # of squares possible is 64+49+36+25+16+9+4+1=204.

P.S. Not a GMAT questions
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21 Sep 2010, 17:59
great explanation Bunuel, I would have never thought of it that way.
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31 May 2014, 13:05
Unable to understand highlighted part:

Bunuel wrote:
Another way:

2. The number of squares on 8X8 chessboard is
a) 204
b) 220
c) 240
d) 210

# of squares with are of 1*1=1 is 8*8=64; -- this one is logical 8 rows 8 columns total 64, 1 unit area squares.
# of squares with are of 2*2=4 is 7*7=49; -- but 7x7 why and how to visualize this on board.
# of squares with are of 3*3=9 is 6*6=36;
...
# of squares with are of 8*8=64 is 1*1=1;

Total # of squares possible is 64+49+36+25+16+9+4+1=204.

P.S. Not a GMAT questions

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31 May 2014, 16:09
PiyushK wrote:
Unable to understand highlighted part:

Bunuel wrote:
Another way:

2. The number of squares on 8X8 chessboard is
a) 204
b) 220
c) 240
d) 210

# of squares with are of 1*1=1 is 8*8=64; -- this one is logical 8 rows 8 columns total 64, 1 unit area squares.
# of squares with are of 2*2=4 is 7*7=49; -- but 7x7 why and how to visualize this on board.
# of squares with are of 3*3=9 is 6*6=36;
...
# of squares with are of 8*8=64 is 1*1=1;

Total # of squares possible is 64+49+36+25+16+9+4+1=204.

P.S. Not a GMAT questions

# of squares with are of 2*2=4 is 7*7=49;
Rows: {1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8} --> 7 positions.
Columns: {1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8} --> 7 positions.

# of squares with are of 3*3=9 is 6*6=36;
Rows: {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 8} --> 6 positions.
Columns: {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 8}--> 6 positions.

Not a GMAT question. So, you can ignore.
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Re: The number of rectangles that can be formed on a 8X8 chessbo  [#permalink]

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03 Oct 2017, 14:13
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Re: The number of rectangles that can be formed on a 8X8 chessbo &nbs [#permalink] 03 Oct 2017, 14:13
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