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Math Help - Need Help

  1. #1
    Super Member malaygoel's Avatar
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    Need Help

    Please solve the following question:
    If a cube is cut into finite number of smaller cubes, prove that at least two of them must be of same size.
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  2. #2
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    Quote Originally Posted by malaygoel
    Please solve the following question:
    If a cube is cut into finite number of smaller cubes, prove that at least two of them must be of same size.
    What about,
    3^3+4^3+5^3=6^3
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  3. #3
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    Quote Originally Posted by ThePerfectHacker
    What about,
    3^3+4^3+5^3=6^3
    You cannot form a cube of side 6 from two cubes of sides 3 and 4 and 5.
    since if one cube were adjacent to one of the others at least one dimension
    of the resulting solid must be 7 or greater.

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack
    You cannot form a cube of side 6 from two cubes of sides 3 and 4.

    RonL
    I think I am missing something here.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    I think I am missing something here.
    We are talking about cutting a cube of side 6 into smaller cubes. That
    means that if two of the smaller cubes were adjacent before disassembly
    of the big cube then the sum of their sides must be <=6 otherwise they
    cannot have fitted into the cube of side 6.

    RonL
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  6. #6
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    Hello, ThePerfectHacker!

    I think I am missing something here.
    You took a six-inch cube and cut out a 5-inch-cube.

    Then you took the remaining volume, diced it into 91 one-inch cubes,
    and reassembled them into a 3-inch cube and a 4-inch cube.
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  7. #7
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    I see what you are saying.
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  8. #8
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Soroban
    Hello, ThePerfectHacker!


    You took a six-inch cube and cut out a 5-inch-cube.

    Then you took the remaining volume, diced it into 91 one-inch cubes,
    and reassembled them into a 3-inch cube and a 4-inch cube.
    prove that there will be atleast two cubes of same size
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  9. #9
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    Can I rephraze problem az,
    \sum_{k=1}^n x_k^3=y^3
    Where, x_i< x_j iff i<j.
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  10. #10
    Super Member malaygoel's Avatar
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    Quote Originally Posted by ThePerfectHacker
    Can I rephraze problem az,
    \sum_{k=1}^n x_k^3=y^3
    Where, x_i< x_j iff i<j.
    Keep in mind that sum of some x's should be y(edge of cube)
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  11. #11
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    Quote Originally Posted by malaygoel
    Keep in mind that sum of some x's should be y(edge of cube)
    x_1+x_2+...+x_n=y
    With,
    (x_1)^3+(x_2)^3+...+(x_n)^3=y^3
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  12. #12
    Super Member malaygoel's Avatar
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    Quote Originally Posted by ThePerfectHacker
    x_1+x_2+...+x_n=y
    With,
    (x_1)^3+(x_2)^3+...+(x_n)^3=y^3
    I think you have made the wrong equations
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