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Puzzle – How many rectangles on a chess board? – Solutions

April 21, 2007 pm30 10:07 pm

This is the place for solutions to the rectangles on the chess board problem. The original problem is here.

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84 Comments leave one →
  1. David permalink
    April 29, 2007 am30 6:48 am 6:48 am

    A rectangle on a chess board is bounded by two vertical grid lines and two horizontal grid lines. Since a chess board has nine vertical and nine horizontal grid lines, the number of rectangles is C(9,2) * C(9,2) = 1296.

    • soumya permalink
      July 2, 2009 am31 12:26 am 12:26 am

      ACTUALLY THERE ARE 8 HORIZONTAL AND VERTICAL LINE

      • August 29, 2009 am31 1:19 am 1:19 am

        count it, there are 9, since there are 8 boxes

      • allah permalink
        December 6, 2009 pm31 2:19 pm 2:19 pm

        soumya, you are very stupid

        • googed permalink
          May 13, 2010 am31 6:35 am 6:35 am

          allah has spoken!

        • smokey permalink
          February 9, 2011 pm28 1:20 pm 1:20 pm

          u are very very stupid

      • soumya's bf permalink
        November 9, 2010 am30 4:14 am 4:14 am

        soumya i dint know u r so dumb.
        you lack basic knowledge in mathematics.
        stupid soumya!!

        • Anonymous permalink
          May 28, 2011 pm31 2:05 pm 2:05 pm

          a** h**** soumya

        • soumya's latest lover permalink
          July 6, 2011 am31 2:23 am 2:23 am

          hehehe bas naam hi kaafi hai

        • January 19, 2012 pm31 1:49 pm 1:49 pm

          be quite son of b****

      • Anonymous permalink
        May 10, 2012 pm31 7:07 pm 7:07 pm

        Seriously? COUNT THEM!!!!

    • rakshit permalink
      January 19, 2010 pm31 2:46 pm 2:46 pm

      u r right

    • Raj permalink
      January 17, 2011 am31 6:52 am 6:52 am

      David has given the most easy and effective solution, what for all these stupid and confusing discussion……………….

    • manish karki permalink
      January 21, 2011 am31 6:36 am 6:36 am

      hi dosto

    • Prabhu Dhev permalink
      July 1, 2011 am31 3:20 am 3:20 am

      Lovely answer David! The problem puzzled me for a long time.

      • kshitij permalink
        August 19, 2011 am31 1:07 am 1:07 am

        thanx david!

      • Anonymous permalink
        August 31, 2011 am31 5:09 am 5:09 am

        Also, an alternative soln is summation of cubes from 1 to 8 ie (1+8+27+64+125+216+343+512 =1296)

        -Jyoti

        • dinesh permalink
          November 12, 2011 am30 6:07 am 6:07 am

          no there is not 1296 rectangle there is 1296-204=1092 rectangle
          N3-N2 where n=1 2 3……8
          1+8+27+64+125+216+343+512-(1+4+9+16+25+36+49+64)=1092

        • Mercy permalink
          January 6, 2012 am31 12:51 am 12:51 am

          I didn’t understand your logic of summation. Why did you do this?

        • Anonymous permalink
          May 10, 2012 pm31 7:12 pm 7:12 pm

          You’re right, adding the cube numbers. :)

    • Anonymous permalink
      August 24, 2011 am31 11:58 am 11:58 am

      thanx david guetta for the answer

    • Anonymous permalink
      September 16, 2012 am30 8:05 am 8:05 am

      great david !!!!

    • Anonymous permalink
      February 27, 2013 pm28 1:42 pm 1:42 pm

      How do you work out c(9*2) ?

    • Anonymous permalink
      February 27, 2013 pm28 1:44 pm 1:44 pm

      how do you do c(9^2) ?

      • March 3, 2013 pm31 11:08 pm 11:08 pm

        c(9,2) would be combinations of nine things, taken two at a time. (9 x 8) / (1 x 2) = 36. How do you get it? There are 9 lines (dividing 8 rows). Take any two of them…

  2. April 30, 2007 am30 1:17 am 1:17 am

    Very cool – a brand new approach for me. I have a few others, but this puzzle doesn’t seem to have caught my usual puzzlers’ imaginations. I was waiting for more response.

  3. May 6, 2007 am31 4:39 am 4:39 am

    You can do it the same way as the square problem. The nth bottommost, mth rightmost square is the northwest corner of mn rectangles. The nth row has northwest corners of (1 + … + 8)n = 36n rectangles, so in total there are 36(1 + … + 8 ) = 36^2 = 1296 rectangles.

    And in general, on an m*n board there are mn(m+1)(n+1)/4 squares, the same answer you’d get by generalizing David’s approach but by a different method.

  4. May 6, 2007 am31 4:55 am 4:55 am

    My first solution was by dimensions. Consider a 3×3 chessboard.

    There are 9 1×1′s, 6 2×1′s, 3 3×1′s
    There are 6 1×2′s, 4 2×2′s, 2 3×2′s
    There are 3 1×3′s, 2 2×3′s, 1 3×3

    Trotting out the LaTeX, that’s
    \overset{3} \sum _{n=1}{3n} + \overset{3} \sum _{n=1}{2n} + \overset{3} \sum _{n=1}{1n} , which equals

    3( \overset{3} \sum_{n=1}{n} ) + 2( \overset{3} \sum _{n=1}{2} ) + 1( \overset{3} \sum _{n=1}{n} ) , which can be factored as

    (\overset{3}  \sum_{n=1}{n}) \times (\overset{3}  \sum_{n=1}{n})

    From there we reach
    \biggl(\frac{n(n+1)}{2} \biggr)^2 = \frac{(n^2 + n)^2}{4} , or,
    as you’ve generalized for any m x n rectangle,
    \frac{(m^2 + m) \times (n^2 +n)}{4}

    • Anonymous permalink
      January 27, 2012 pm31 5:45 pm 5:45 pm

      soumya…..u r really really stupid….
      n jd2718 ur very very very genious………….
      wt abt triangel gerneralised formula..?

  5. May 9, 2007 am31 3:17 am 3:17 am

    Incidentally, this generalizes to n-dimensional boards. A board of size m1*m2*…*m(n) has m1(m1+1)…m(n)(m(n)+1)/2^n n-dimensional rectangular polytopes.

  6. butt permalink
    February 13, 2008 am29 3:54 am 3:54 am

    204 squares

  7. sylvia permalink
    June 5, 2008 pm30 1:00 pm 1:00 pm

    Why are ther 204 squares in a chess board?

  8. June 6, 2008 am30 4:49 am 4:49 am

    Because we count all the different sized squares, not just the little ones.

  9. lukey permalink
    September 13, 2008 pm30 6:55 pm 6:55 pm

    i make it 1365

    • yash permalink
      May 30, 2009 am31 5:48 am 5:48 am

      there are 1296 rectangle in an chess board

  10. josh permalink
    October 7, 2008 pm31 9:01 pm 9:01 pm

    i worked out a way an i got 1278

  11. maa permalink
    October 26, 2008 pm31 2:54 pm 2:54 pm

    there are 1296 different rectangles in a chessboard..

  12. amber permalink
    November 16, 2008 pm30 10:34 pm 10:34 pm

    how many squares and rectangles aee in a chess board…………………………………………………………………………………………………………………………………………………………………………

    • Anonymous permalink
      February 27, 2013 pm28 1:37 pm 1:37 pm

      1296 rectangles + 204 squares

  13. November 20, 2008 pm30 11:11 pm 11:11 pm

    The easiest way to work this out, is to see a patern. So, see how many sqaures and rectangles there are in a 1 by 1 grid and add up total. Answer is 1. do that for 2,3,4,5,6,7 and 8. It may take a while but you will get the answer in the end.

    • August 29, 2009 am31 1:17 am 1:17 am

      too slow…for my homework, i get a third of a page to explain it

  14. Krishnan permalink
    December 1, 2008 pm31 4:35 pm 4:35 pm

    In a standard chessboard, you have 9 horizontal and 9 vertical lines. Choose any two horizontal and any two vertical lines to form a rectangle. Both of them can be chosen in 9c2 ways. So the total number of rectangles is 9c2 * 9c2, one for horiznotal, and one for vertical. Simple

    • LOl!! permalink
      August 25, 2012 pm31 2:19 pm 2:19 pm

      can u explain the same for square…???

  15. December 1, 2008 pm31 4:46 pm 4:46 pm

    That’s gorgeous, thank you.

    And, it generalizes eg how many boxes in a Rubik’s cube? (3 x 3 x 3)

  16. j'lho;[o8hy permalink
    January 26, 2009 am31 5:45 am 5:45 am

    dumm puzle

  17. hema permalink
    March 16, 2009 am31 2:16 am 2:16 am

    How many different rectangles which are not squares can be found on a chess board? ( a chess board has 8 rows and 8 columns of squares. a rectangle on the board is a collection of squares that form a rectangular piece of the whole board. Two rectangles are different if they have different sets of squares. A rectangle with an unequal number of rows and columns is not square.)

  18. March 16, 2009 am31 7:50 am 7:50 am

    Hi hema, your question is part of the current CUNY math challenge – it wouldn’t be fair for me to answer.

  19. Rachael permalink
    April 21, 2009 pm30 11:32 pm 11:32 pm

    I started from a 1*1 chess board and worked my way up to 1 4*4 chess board before I spotted a pattern.
    1*1 = 1 square ‘a’
    2*2 = 9 squares ‘b’
    3*3 = 36 squares ‘c’
    4*4 = 100 sqauers ‘d’

    Let us call the number that is beign squared ‘n’.
    I saw that the differences between n each time was going up by 1. For example:
    number of squares on ‘a’ = 1
    = 1*1
    number of squares on ‘b’ = 9
    = 3*3
    number of squares on ‘c’ = 36
    = 6*6
    number of squares on ‘d’ = 100
    = 10*10
    The sequence is 1, 3, 6, 10 and therefore the differences are 2,3,4, and so I carried on with the sequence by adding 1 to the differences until I worked out that the total numbrer of rectangles on a chess board must be:
    28*28 = 784
    The number was lower than I expected, but I have yet to test out the hypothesis.

  20. keerthana permalink
    May 10, 2009 am31 3:16 am 3:16 am

    1296

  21. May 30, 2009 am31 5:50 am 5:50 am

    there are 1296 rec. on chess board

  22. May 30, 2009 am31 5:51 am 5:51 am

    there r 1296 rec. on chess board

  23. May 30, 2009 am31 5:52 am 5:52 am

    and there are 204 squares

  24. June 14, 2009 am30 10:15 am 10:15 am

    YAA Adding this to my bookmarks. Thank You

  25. June 14, 2009 am30 10:22 am 10:22 am

    YAA Adding this to my bookmarks. Thank You

  26. Ashish permalink
    July 2, 2009 am31 3:57 am 3:57 am

    there are thousands of rectangles in a chess boards…
    and 1296 is the number of squares..
    but if u concentrate the number of rectangles…then it wud b lot more..

    • Anonymous permalink
      May 10, 2012 pm31 7:10 pm 7:10 pm

      No, Ashish. There are 204 sqaures and 1092 rectangles and you add them together to get 1296.

  27. jason permalink
    July 10, 2009 pm31 11:35 pm 11:35 pm

    there is 1…the board itself

    • July 11, 2009 pm31 1:12 pm 1:12 pm

      Absolutely. But there are others as well.

  28. Struggler permalink
    October 21, 2009 pm31 6:59 pm 6:59 pm

    What does 9c2 mean? Im really confused and trying figure this problem out but it is hard not understanding what everything means.. Can someone help, please!

    • October 22, 2009 pm31 11:02 pm 11:02 pm

      9C2 is combinations of 9 things taken 2 at a time. For example, how many ways could you choose two books out of nine possible? We also write C(9,2) or \binom{9}{2} or _{9} C_{2}

  29. Vaibhav Chicholikar permalink
    July 20, 2010 am31 6:37 am 6:37 am

    there are 1296 – 204 = 1092 rectangles..

    • Sunil permalink
      August 3, 2010 pm31 2:04 pm 2:04 pm

      Finally the Answer I have been looking for ..
      Out of those 9c2 * 9c2 quadrilaterals, some are squares…So no.of rectangles = no.of quadrilaterals – no.of Squares

  30. October 1, 2010 am31 8:49 am 8:49 am

    1296 rectangles in a chess board

  31. Ashish permalink
    December 3, 2010 am31 10:42 am 10:42 am

    Hey pleas say hw its 1092 rectangles…explain

    • himu permalink
      December 15, 2010 pm31 3:06 pm 3:06 pm

      all square are rectangles but all rectangles are not square so the answer is not 1092 it is 1926

  32. January 21, 2011 am31 10:30 am 10:30 am

    there is a nice solution to the how many squares or rectangles are on a chessboard problem here
    http://puzzles.nigelcoldwell.co.uk/twentyseven.htm
    with diagrams

  33. Anonymous permalink
    March 28, 2011 am31 10:59 am 10:59 am

    hello champions

  34. June 25, 2011 am30 9:15 am 9:15 am

    i think dat problem is related to combination. actually the wright answer is 1296.

  35. gayu permalink
    December 9, 2011 am31 9:43 am 9:43 am

    thanks david u r so brilliant.

  36. January 19, 2012 pm31 1:46 pm 1:46 pm

    hi stop rude shutt your bitch fat gob

  37. January 19, 2012 pm31 1:48 pm 1:48 pm

    there are 1926

  38. Anonymous permalink
    May 10, 2012 pm31 7:08 pm 7:08 pm

    I figured it out in grade 3, honestly. I did it the long way but found a few patterns, and I got 1296 or 32 squared. I am very proud. :)

  39. Anonymous permalink
    May 10, 2012 pm31 7:11 pm 7:11 pm

    I found patterns when I figured it out in grade 3 so I didn’t take, like two years. I am NOT lying.

  40. Anonymous permalink
    July 21, 2012 am31 5:54 am 5:54 am

    salo bakwas kr rhe ho ans kyo ni dete

  41. Anonymous permalink
    July 21, 2012 am31 5:56 am 5:56 am

    288

  42. Syd permalink
    September 10, 2012 pm30 4:41 pm 4:41 pm

    64 squares
    512 rectangles

  43. Anonymous permalink
    September 12, 2012 pm30 3:10 pm 3:10 pm

    1296
    9C2 *9C2

  44. Anonymous permalink
    September 22, 2012 am30 12:39 am 12:39 am

    1296

  45. Anonymous permalink
    March 6, 2013 pm31 1:20 pm 1:20 pm

    I’m stuck and this didn’t help at all

  46. prince permalink
    August 19, 2013 am31 10:34 am 10:34 am

    there is 1092 rectangles in chess board

  47. prince permalink
    August 19, 2013 am31 10:36 am 10:36 am

    and there is 204 square in chess board

  48. sajeevan m. narayanan permalink
    April 1, 2014 am30 1:56 am 1:56 am

    respect sirs, i got there are 1296 rectangles on a chess board. but i think it is 1 less means 1295. can you please give me the reply? my email id= mail4sajee@gmail.com

  49. April 16, 2014 am30 2:08 am 2:08 am

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