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A recess in a cube

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  • gulland68
    Is the following intuition of mine, in the final sentence of the following paragraph, determinable by trivial mathematical means? When a cubic structure is
    Message 1 of 2 , May 7, 2004
      Is the following intuition of mine, in the final sentence of the
      following paragraph, determinable by trivial mathematical means?

      When a cubic structure is subtracted from a larger cube composed of
      cubic blocks, the blocks remaining from the larger cube cannot be
      reassembled into a cube, since if the subtracted cube is removed in
      such a way that it an the larger cube share a common vertex, there
      are invariably two sides of the remains of the host cube that are
      rectangular if the third is taken to be a square base. Howeverone
      rearranges the blocks into squares there must be some blocks
      remaining that can only be arranged as a rectangle or not grouped
      together as a quadrangle at all.

      Thanks,
      Tom
    • richard_heylen
      ... I think that the fact that a^3 + b^3 c^3 for integers makes it seem reasonable to me that this assertion is true. I can t however, off the top of my
      Message 2 of 2 , May 7, 2004
        --- In primenumbers@yahoogroups.com, "gulland68" <tmgulland@h...>
        wrote:
        > When a cubic structure is subtracted from a larger cube composed of
        > cubic blocks, the blocks remaining from the larger cube cannot be
        > reassembled into a cube

        I think that the fact that a^3 + b^3 <> c^3 for integers makes it
        seem reasonable to me that this assertion is true. I can't however,
        off the top of my head, rule out the cases in which the ratios of the
        lengths of the cubic blocks are irrational or in which two or more of
        the cubes comprise an infinite number of cubic pieces.

        , since if the subtracted cube is removed in
        > such a way that it an the larger cube share a common vertex, there
        > are invariably two sides of the remains of the host cube that are
        > rectangular if the third is taken to be a square base. Howeverone
        > rearranges the blocks into squares there must be some blocks
        > remaining that can only be arranged as a rectangle or not grouped
        > together as a quadrangle at all.

        I don't fully understand what you mean and what I think I understand,
        I think is wrong. It might be instructive to consider a cube divided
        into 43944934864044921875 equal cubic pieces. You can then remove a
        cube with 1214928 pieces along an edge and rearrange the remainder to
        form a cube with 3480205 pieces along an edge. You then only have 2
        cubic pieces left over!

        Richard
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