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Math Help - How to fill in this table?

  1. #1
    Junior Member
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    Vadodara,Gujarat,India
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    Question How to fill in this table?

    Hope it belongs to the right section....Following is a table.Here is the question:
    Fill in the remaining areas by real no.s such that the sum of the numbers in any three neighboring squares is constant and sum of all no.s is 210.

    |_|_|26|_|_|_|12|_|_|_|

    Is it only by trial & error?
    Sorry,dont know how to make tables
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  2. #2
    Super Member
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    Quote Originally Posted by anshulbshah View Post
    Hope it belongs to the right section....Following is a table.Here is the question:
    Fill in the remaining areas by real no.s such that the sum of the numbers in any three neighboring squares is constant and sum of all no.s is 210.

    |_|_|26|_|_|_|12|_|_|_|

    Is it only by trial & error?
    Sorry,dont know how to make tables
    Filling in the blanks:
    .a.b.26.c.d.e.12.f.g.h.

    Resulting equations:
    a+b+26 = x
    b+26+c=x
    26+c+d=x
    c+d+e=x
    d+e+12=x
    e+12+f=x
    12+f+g=x
    f+g+h=x
    a+b+26+c+d+e+12+f+g+h = 210

    You have nine equations and nine unknowns.

    There are several ways to attack this:

    You can remove the x by subtracting the adjoining pairs.

    a+b+26 = x
    b+26+c=x
    a - c = 0

    26+c+d=x
    c+d+e=x
    26 - e =0

    d+e+12=x
    e+12+f=x
    d - f =0

    12+f+g=x
    f+g+h=x
    12 - h = 0

    That's a start.
    and continue with every other equation
    a+b+26 = x
    26+c+d=x
    a+b = c+d

    26+c+d=x
    d+e+12=x
    26+c = e +12

    etc..

    Or create an augmented matrix and reduce to a triangular matrix.
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  3. #3
    Super Member

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    Lexington, MA (USA)
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    Hello, anshulbshah!

    Fill in the remaining cells with real nnumbers
    so that the sum of three consecutive numbers is constant
    and sum of all numbers is 210.

    . . \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline<br />
\;\;& \;\;& 26 & \;\; & \;\; & \;\; & 12 & \;\; & \;\;& \;\; \\ \hline \end{array}
    We are given: . \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline<br />
a & b & 26 & c & d & e & 12 & f & g & h \\ \hline \end{array}

    We have: . a+b+26 \:=\:b+26+c \quad\Rightarrow\quad a \:=\:c
    We have: . c+d+e \:=\:d+e+12 \quad\Rightarrow\quad c \:=\:12
    We have: . 12+f+g \:=\:f+g+h \quad\Rightarrow\quad h \:=\: 12
    . . Hence: . a \:=\:c\:=\:h \:=\:12

    Answer (so far): . \begin{array} {|c|c|c|c|c|c|c|c|c|c|} \hline 12 & b & 26 & 12 & d & e & 12 & f & g & 12 \\ \hline \end{array}


    We have: . 26 + 12 + d \:=\:12+d+e \quad\Rightarrow\quad e \:=\:26
    We have: . e+12+f \:=\:12+f+g \quad\Rightarrow\quad e \:=\:g
    . . Hence: . e \:=\:g\:=\:26

    Answer (so far): . \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 12 & b & 26 & 12 & d & 26 & 12 & f & 26 & 12 \\ \hline \end{array}


    We have: . b+26+12 \:=\:26 + 12 + d \quad\Rightarrow\quad b \:=\:d
    We have: . d+e+12 \:=\:e+12+f \quad\Rightarrow\quad d \:=\:f
    . . Hence: . b \:=\:d\:=\:f


    The sum of all numbers is 210.

    . . 12 + b + 26 + 12 + d + 26 + 12 + f + 26 + 12 \:=\:210

    . . b+d+f + 126 \:=\:210 \quad\Rightarrow\quad b+d+f \:=\:84

    Since b=d=f, we have: . 3b \:=\:84 \quad\Rightarrow\quad b \:=\:28

    . . Hence: . b\:=\:d\:=\:f\:=\:28


    Answer: . {\color{blue}\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 12 & 28 & 26 & 12 & 28 & 26 & 12 & 28 & 26 & 12 \\ \hline \end{array}}

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