Re: A recess in a cube
- --- In firstname.lastname@example.org, "gulland68" <tmgulland@h...>
> When a cubic structure is subtracted from a larger cube composed ofI think that the fact that a^3 + b^3 <> c^3 for integers makes it
> cubic blocks, the blocks remaining from the larger cube cannot be
> reassembled into a cube
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, thereI don't fully understand what you mean and what I think I understand,
> 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 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!