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Arrange the numbers 1 through 10 in the shape of a pyramid

  1. Feb 3, 2004 #1
    Arrange the numbers 1 through 10 in the shape of a pyramid (1 number at the peak, 2 numbers in the next row down, 3 in the next row, and 4 in the bottom row). The numbers must be arranged such that each number is equal to the difference of the two numbers directly below it. For example, if 5 and 1 are next to each other in the same row, then 4 must be placed on top of them.
     
  2. jcsd
  3. Feb 3, 2004 #2
    3
    7 4
    2 9 5
    8 10 1 6
    It represents a pyramid, though I'm unable to give to it a pyramidal shape
     
  4. Feb 3, 2004 #3

    NateTG

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    Science Advisor
    Homework Helper

    It looks good, but if you're reall picky you can use the code tag for fixed spacing

    Code (Text):

       3
      7 4
     2 9 5
    8 A 1 6
     
    I wonder if the solution is unique. Apparently not:

    Code (Text):

       3
      2 5
     7 9 4
    8 1 A 6
     
     
    Last edited: Feb 3, 2004
  5. Feb 3, 2004 #4

    chroot

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    Staff Emeritus
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    Gold Member

    There are eight solutions:

    {3, 2, 5, 7, 9, 4, 8, 1, 10, 6},
    {3, 4, 7, 5, 9, 2, 6, 1, 10, 8},
    {3, 5, 2, 4, 9, 7, 6, 10, 1, 8},
    {3, 7, 4, 2, 9, 5, 8, 10, 1, 6},
    {4, 1, 5, 6, 7, 2, 9, 3, 10, 8},
    {4, 2, 6, 5, 7, 1, 8, 3, 10, 9},
    {4, 5, 1, 2, 7, 6, 8, 10, 3, 9},
    {4, 6, 2, 1, 7, 5, 9, 10, 3, 8}

    - Warren
     
  6. Feb 4, 2004 #5
    For 1 level, there are 4 solutions:

    2 / 1 3
    2 / 3 1
    1 / 2 3
    1 / 3 2

    3 always appears on the bottom.

    For 2 levels, there are 6 solutions:

    2 / 3 5 / 4 1 6
    2 / 5 3 / 6 1 4
    1 / 3 4 / 5 2 6
    1 / 4 3 / 6 2 5
    3 / 4 1 / 2 6 5
    3 / 1 4 / 5 6 2

    6 always appears on the bottom. 5 and 4 appear on the bottom two levels, with only one of these at the highest.

    I trust there are no more than 8 solutions for 3 levels:

    3 / 4 7 / 5 9 2 / 6 1 A 8
    3 / 7 4 / 2 9 5 / 8 A 1 6
    3 / 2 5 / 7 9 4 / 8 1 A 6
    3 / 5 2 / 4 9 7 / 6 A 1 8
    4 / 2 6 / 5 7 1 / 8 3 A 9
    4 / 6 2 / 1 7 5 / 9 A 3 8
    4 / 1 5 / 6 7 2 / 9 3 A 8
    4 / 5 1 / 2 7 6 / 8 A 3 9

    10 always appears on the bottom. 8 and 9 appear on the bottom two levels. (In fact 8 is always on the bottom.) 5, 6, and 7 appear on the bottom three levels, with only one of these at the highest.

    What a great way to waste time...
     
  7. Feb 4, 2004 #6
    meteor with the point, good work chroot and davilla!
     
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