Posers and Puzzles

Posers and Puzzles

  1. Joined
    25 Aug '06
    Moves
    0
    03 Dec '07 18:521 edit
    A matrix of numbers is created by the following procedure. Start with an empty rectangular matrix of any dimensions. Then draw a "snake" of 1's from the SW corner to the NE corner. At each step the snake goes only up or right. Complete the matrix by the following rule: if you have four numbers in the matrix, arranged like this -

    a b
    c d

    then ad-bc should equal 1.

    See an example here: http://www.2send.us/uploads/92b5e9b431.gif

    Prove: In every matrix created by the method just explained, all the numbers are integers.
  2. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
    Moves
    28719
    03 Dec '07 19:161 edit
    Originally posted by David113
    A matrix of numbers is created by the following procedure. Start with an empty rectangular matrix of any dimensions. Then draw a "snake" of 1's from the SW corner to the NE corner. At each step the snake goes only up or right. Complete the matrix by the following rule: if you have four numbers in the matrix, arranged like this -

    a b
    c d

    then ad-bc ...[text shortened]...

    Prove: In every matrix created by the method just explained, all the numbers are integers.
    Just to be clear, is the result that the determinant of any matrix constructed this way will always be 1? And the problem is to prove this result?

    EDIT: Oops, I get it now...insert the 1's first, then use the rule to fill in the blanks, then prove that all the blanks must be filled with integers.