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Math Help - Puzzle-like problem

  1. #1
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    Puzzle-like problem

    Hi,

    On a square game-board that is divided into n rows of n squares each, k of these squares lie along the boundary of the game-board. Which of the following is a possible value of k?

    A. 10
    B. 25
    C. 34
    D. 42
    E. 52

    I have 25 (B). I don't have much reasoning behind my choice. Not sure if it requires much reasoning anyway. It mentions squares and the only number that is a perfect square is 25.
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  2. #2
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    Quote Originally Posted by Hellbent View Post
    Hi,

    On a square game-board that is divided into n rows of n squares each, k of these squares lie along the boundary of the game-board. Which of the following is a possible value of k?

    A. 10
    B. 25
    C. 34
    D. 42
    E. 52

    I have 25 (B). I don't have much reasoning behind my choice. Not sure if it requires much reasoning anyway. It mentions squares and the only number that is a perfect square is 25.
    The perimeter is k, which is 2n+2(n-2)=4n-4

    10=14-4 but 14 is not a multiple of 4
    25=29-4
    34=38-4
    42=46-4
    52=56-4

    56 is the only integer multiple of 4.
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  3. #3
    Super Member Quacky's Avatar
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    Edit: Nevermind, too late. Seriously, why do I even bother trying?
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  4. #4
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    Quote Originally Posted by Quacky View Post
    Edit: Nevermind, too late. Seriously, why do I even bother trying?
    You may well have said it in a much more eloquent way!
    Anyways... I was playing chess a couple hours ago
    so I had an unfair advantage
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  5. #5
    Super Member Quacky's Avatar
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    Quote Originally Posted by Archie Meade View Post
    You may well have said it in a much more eloquent way!
    Anyways... I was playing chess a couple hours ago
    so I had an unfair advantage
    Yup. Dirty cheat. *Incomprehensible comment about how my reasoning just complicated the question even further*
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  6. #6
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    Quote Originally Posted by Archie Meade View Post
    The perimeter is k, which is 2n+2(n-2)=4n-4

    10=14-4 but 14 is not a multiple of 4
    25=29-4
    34=38-4
    42=46-4
    52=56-4

    56 is the only integer multiple of 4.
    Yeah, I never had faith in my approach. I don't follow this: 2n+2(n-2)=4n-4.

    I think I follow the rest: a square has 4 sides, so you replaced n with each answer choice in the formula and checked which numbers would give a multiple of 4. Am I following this correctly?
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  7. #7
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    Quote Originally Posted by Hellbent View Post
    Yeah, I never had faith in my approach. I don't follow this: 2n+2(n-2)=4n-4.

    I think I follow the rest: a square has 4 sides, so you replaced n with each answer choice in the formula and checked which numbers would give a multiple of 4. Am I following this correctly?
    It's different to finding a perimeter.
    We add all four side lengths there.

    The problem here is that if you add up all the squares on the 2 horizontal sides, that's 2n.
    In adding the squares left to be counted on the vertical sides,
    notice that we've already accounted for the 4 squares at the corners!
    So the amount of squares remaining to be counted is (n-2)+(n-2).
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  8. #8
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    Hello, Hellbent!

    Another way to count the perimeter . . .


    \text{On an }n\times n\text{ chessboard, }k\text{ squares lie on the  boundary of the board.}
    \text}{Which of the following is a possible value of }k?

    . . (A)\; 10 \qquad (B)\;25 \qquad (C)\;34 \qquad (D)\;42 \qquad (E)\;52
    Code:
    
          : - -  n-1  - - :
          ♥ ♥ ♥ ♥ . . . ♥ ♥ ♠ -
        - ♣                 ♠ :
        : ♣                 ♠ :
        : .                 ♠ :
        : .                 .n-1
       n-1.                 . :
        : ♣                 . :
        : ♣                 ♠ :
        : ♣                 ♠ -
        - ♣ ◊ ◊ . . . ◊ ◊ ◊ ◊
            : - -  n-1  - - :

    Each of the four sides has n-1 squares.

    Hence, there are: . 4(n-1) boundary squares, a multiple of 4.

    The only multiple of 4 is: . (E)\;52

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  9. #9
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    Here is the way I would do this problem on the SAT:

    First I draw some pictures.

    **
    **

    ***
    ***
    ***

    ****
    ****
    ****
    ****

    Above are pictures for n=2,3,4
    The corresponding values for k are as follows:

    n=2 k=4
    n=3 k=8
    n=4 k=12

    Draw as many values of n as you need to until you realize that the possible values of k are all positive multiples of 4.

    Choice (E) is the only multiple of 4.
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