Results 1 to 4 of 4

Math Help - Matrix Puzzle

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    12

    Matrix Puzzle

    Suppose that we have a 3x3 matrix such that the sum of each row and that of each column is equal to the same non-negative integer s. If each entry is known to be a non-negative integer and same number can be used more than once in each row or in each column, how many such 3x3 matrices are there?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,663
    Thanks
    603
    Hello, hastings!

    Are they any other restrictions to the numbers?
    As stated, the problem is rather silly.


    Suppose that we have a 3x3 matrix such that the sum of each row
    and that of each column is equal to the same non-negative integer S.
    If each entry is known to be a non-negative integer
    and same number can be used more than once in each row or in each column,
    how many such 3x3 matrices are there? . . {\color{blue}\text{Answer: }\:\infty}

    Since numbers can be repeated, the problem is trivial.

    . . A solution is: . \left[\begin{array}{ccc} n & n & n \\ n & n & n \\ n& n & n\end{array}\right] . for any n \in I^+



    Using three different integers: a,b,c

    . . a solution is: . \left[\begin{array}{ccc}a & b & c \\ b & c & a \\ c & a & b\end{array}\right] . for any a,b,c \in I^+



    Even if the numbers must be consecutive integers,

    . . a solution is: . \left[\begin{array}{ccc} a+7 &a+2 & a+3 \\ a & a+4 & a+8 \\ a+5 & a+2 & a+1 \end{array}\right] . for any a \in I^+

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2008
    Posts
    12
    Hello soroban,

    This is what i found, too, but the person who asked me this puzzle said that the answer was different. Anyways, if no repetition is allowed, is there a formula for the number of such matrices?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Soroban View Post
    Hello, hastings!

    Are they any other restrictions to the numbers?
    As stated, the problem is rather silly.



    Since numbers can be repeated, the problem is trivial.

    . . A solution is: . \left[\begin{array}{ccc} n & n & n \\ n & n & n \\ n& n & n\end{array}\right] . for any n \in I^+



    Using three different integers: a,b,c

    . . a solution is: . \left[\begin{array}{ccc}a & b & c \\ b & c & a \\ c & a & b\end{array}\right] . for any a,b,c \in I^+



    Even if the numbers must be consecutive integers,

    . . a solution is: . \left[\begin{array}{ccc} a+7 &a+2 & a+3 \\ a & a+4 & a+8 \\ a+5 & a+2 & a+1 \end{array}\right] . for any a \in I^+

    The question wants to know for a fixed s \in \mathbb{N} how many N(s) distinct matrices satisfying the condition exist.

    The obvious results are N(1)=0, \ N(2)=0,\ N(3)=1,\ N(4)=6.

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with a puzzle
    Posted in the Algebra Forum
    Replies: 3
    Last Post: December 13th 2011, 02:00 PM
  2. Puzzle
    Posted in the Math Challenge Problems Forum
    Replies: 3
    Last Post: December 22nd 2006, 03:38 PM
  3. A puzzle
    Posted in the Math Challenge Problems Forum
    Replies: 4
    Last Post: December 9th 2006, 07:40 PM
  4. another puzzle
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 7th 2006, 07:35 PM
  5. another puzzle
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: March 7th 2006, 07:33 PM

Search Tags


/mathhelpforum @mathhelpforum