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Math Help - Brainy 2

  1. #1
    Member u2_wa's Avatar
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    Brainy 2

    Fit digits 1 to 8 in such a way that no two consecutive numbers are next to each other.
    An example is also attached.
    Please count your attempts!
    Attached Thumbnails Attached Thumbnails Brainy 2-tricky.jpg   Brainy 2-tricky1.jpg  
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  2. #2
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    Hello, u2_wa!

    This is an excellent puzzle!


    Place the digits 1 to 8 so that no two consecutive numbers are adjacent.
    ("Adjacent" numbers are in cells that share a side or a vertex.)
    Code:
              *---*
              |   |
          *---*---*---*
          |   |   |   |
          *---*---*---*
          |   |   |   |
          *---*---*---*
              |   |
              *---*

    I solved this while in college (back in the Jurassic Period)
    . . and have a logical solution.

    I'll post it later, after everyone has tried it.

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  3. #3
    A riddle wrapped in an enigma
    masters's Avatar
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    Quote Originally Posted by u2_wa View Post
    Fit digits 1 to 8 in such a way that no two consecutive numbers are next to each other.
    An example is also attached.
    Please count your attempts!
    Hi u2_wa,

    I have no logical approach to this one, but through trial and many errors, I came up with this. I'll 'white' it out, in case others are still struggling and want to solve it.

    Code:
     
        | 2  |
    --------------
    | 6 | 8  | 5 |
    --------------
    | 4 | 1  | 3 |
    --------------
        | 7  |
    Last edited by masters; March 17th 2009 at 08:19 AM.
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  4. #4
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by masters View Post
    Hi u2_wa,

    I have no logical approach to this one, but through trial and many errors, I came up with this. I'll 'white' it out, in case others are still struggling and want to solve it.

    Code:
     
        | 2  |
    --------------
    | 6 | 8  | 5 |
    --------------
    | 4 | 1  | 3 |
    --------------
        | 7  |
    Believe it or not I got that solution on 2nd or 3rd try but unfortunately I was searching for the number of solutions to this problem before I read your posts

    Please count your attempts!
    Uncountable steps after solving the problem in 3rd
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  5. #5
    Member u2_wa's Avatar
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    Quote Originally Posted by ADARSH View Post
    Believe it or not I got that solution on 2nd or 3rd try but unfortunately I was searching for the number of solutions to this problem before I read your posts


    Uncountable steps after solving the problem in 3rd
    Hello ADARSH:
    I also did it in the second attempt, let us see what others do!!
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  6. #6
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    Okay, here's my "logical" approach to this puzzle.

    We have the eight cells labeled a to h.
    There are two "central cells", c and h.
    Code:
              *---*
              | a |
          *---*---*---*
          | b | c | d |
          *---*---*---*
          | e | f | g |
          *---*---*---*
              | h |
              *---*

    There are eight digits to install:
    . . two "end" numbers and six "middle" numbers.

    . . \underbrace{1}_{\text{end}},\underbrace{2,3,4,5,6,  7}_{\text{midde}},\underbrace{8}_{\text{end}}


    Place a middle number in a central cell, say, 4 in cell c.
    Code:
              *---*
              | x |
          *---*---*---*
          | x | 4 | x |
          *---*---*---*
          | x | x | x |
          *---*---*---*
              | h |
              *---*

    We see that its neighbors (3 and 5) cannot be placed
    . . in cells a,b,d,e,f,g.
    So both 3 and 5 cannot be installed.

    Conclusion: the central cells must not contain a middle number.

    Hence, 1 and 8 must go in cells c and f.

    Code:
              *---*
              | a |
          *---*---*---*
          | b | 1 | d |
          *---*---*---*
          | e | 8 | g |
          *---*---*---*
              | h |
              *---*

    2 cannot be in cells a,b,d,e,g \quad\Rightarrow\quad 2 \in h.

    7 cannot be in cells b,d,e,g \quad\Rightarrow\quad 7 \in a.
    Code:
              *---*
              | 7 |
          *---*---*---*
          | b | 1 | d |
          *---*---*---*
          | e | 8 | g |
          *---*---*---*
              | 2 |
              *---*

    3 cannot be in e or g . . . 3 may be in b or d.

    Pick one: say, 3 \in b
    Code:
              *---*
              | 7 |
          *---*---*---*
          | 3 | 1 | d |
          *---*---*---*
          | e | 8 | g |
          *---*---*---*
              | 2 |
              *---*

    6 cannot be in d.

    Also, 6 cannot be in cell e.
    Else 4 and 5 will be in cells d and g (and be adjacent).
    . . Hence: . 6 \in g.
    Code:
              *---*
              | 7 |
          *---*---*---*
          | 3 | 1 | d |
          *---*---*---*
          | e | 8 | 6 |
          *---*---*---*
              | 2 |
              *---*

    4 cannot be in cell e . . . 4 \in d

    And finally: . 5 \in e
    Code:
              *---*
              | 7 |
          *---*---*---*
          | 3 | 1 | 4 |
          *---*---*---*
          | 5 | 8 | 6 |
          *---*---*---*
              | 2 |
              *---*
    
            Solution

    Disregarding rotations and reflections, there is one solution.

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