Results 1 to 5 of 5

Math Help - numbering squares

  1. #1
    shosho
    Guest

    numbering squares

    The numbers 1 to 25 are to be placed on a black and white 5x5 grid so that each number, except 1, is next to (horizontally or vertically), the number one less than itself. eg.



    a) Explain why number 1 must be placed on a black square
    b) Explain why the numbers 1, 3, 5, 7, 9 can not all be placed on one long diagonal
    c) Show that in a completed numbering the sum of the numbers on a long diagonal must be at least 33 and that 33 can occur.

    Any help is appreciated!
    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by shosho View Post
    The numbers 1 to 25 are to be placed on a black and white 5x5 grid so that each number, except 1, is next to (horizontally or vertically), the number one less than itself. eg.



    a) Explain why number 1 must be placed on a black square
    The sequience of squares that have 1, 2, 3, ... , 25 form a path of adjacent
    squareswhich covers the grid such that consecutive numbers are on different
    coloured squares. There are 13 black squares and 12 white squares, so the path
    must start and end on black squares. Therefor 1 is on a black square.

    RonL
    Last edited by CaptainBlack; June 7th 2007 at 02:11 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by shosho View Post
    b) Explain why the numbers 1, 3, 5, 7, 9 can not all be placed on one long diagonal
    Because the chain of squares labled 1,2,3,4,5,6,7,8,9 will divide the grid
    into two seperate parts, so as the path must continue initialy in one of these
    halves, it can never cross again into the other half so all the squares cannot
    be labled according to the rule.

    RonL
    Follow Math Help Forum on Facebook and Google+

  4. #4
    shosho
    Guest
    thankn you very much now i understand those questions

    but could anyone please explain c)?
    That is the hardest question

    thankyuo
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by shosho View Post
    The numbers 1 to 25 are to be placed on a black and white 5x5 grid so that each number, except 1, is next to (horizontally or vertically), the number one less than itself. eg.



    a) Explain why number 1 must be placed on a black square
    b) Explain why the numbers 1, 3, 5, 7, 9 can not all be placed on one long diagonal
    c) Show that in a completed numbering the sum of the numbers on a long diagonal must be at least 33 and that 33 can occur.

    Any help is appreciated!
    Thanks!
    I don't have an answer to c) yet but the approach I will pursue (and you
    might want to think about) is:

    1. find a permitted numbering with a principle diagonal sum of 33.

    2. The largest element on the diagonal if the diagonal sum is 32 or less
    is 16. We know there is no numbering with a maximum of 11 on the diagonal.
    Can we show that with a maximum of 12, 13, 14, 15 and 16 on the diagonal
    we divide the board into two parts that a number path cannot cross. (that
    is we try to extend the argument of part b) to fill in these cases)

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Godel Numbering
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: February 24th 2011, 01:26 PM
  2. Equation numbering
    Posted in the LaTeX Help Forum
    Replies: 0
    Last Post: April 24th 2010, 01:53 AM
  3. Numbering within chapters
    Posted in the LaTeX Help Forum
    Replies: 7
    Last Post: March 7th 2010, 06:35 PM
  4. Godel Numbering
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: January 16th 2010, 02:06 AM
  5. page numbering
    Posted in the LaTeX Help Forum
    Replies: 0
    Last Post: October 24th 2008, 01:17 PM

/mathhelpforum @mathhelpforum