New & old LEGO seqs; production cuts

[Jonathan Post]

Dear Soren,

Thank you for a clear and thoughful answer.

My son (in the room with me now) agrees that baby/childhood play with Duplo
and Lego guided him towards his Math/Computer Science expertise today.

He and I agree, however, that it's a shame that the manufacturer doesn't
make 4-D Lego blocks, or, more properly, the ones that they do make are of
extension along the t-axis only in the trivial way, and cannot be freely
rotated in Minkowski space.

So, I wonder, how well does your approach to counting under symmetry work if
extended to an additional spacial dimension?

I suppose that we can consider an AxBxC 4-D Lego to be actually an AxBxCx1
Lego in Z^4. I think that adjacency is well-defined, all rotations are about
planes, and there is a relationship to 4-D analogues of polyominoes, i.e. to
polyhypercubes.

Best,

Jonathan Vos Post

On 11/2/06, Søren Eilers <eilers at math.ku.dk> wrote:
>
>  Dear Jonathan
>
> Thank you, and yes, I think that LEGO inspires kids to learn about
> fundamental mathematical concepts such as symmetry.
>
> I've also considered adding crossrefs to the sequences you mentioned but
> decided against it.
>
> Regarding A007576: Watson definitely counts a subset of what we count in
> A123818, but as you can see in his paper he only looks at "planar" buildings
> where all blocks could be placed in, say, [0,2]xRxR. This means that one of
> the dimensions of the blocks used is irrelevant, and Watson could just have
> well have counted what we call FLAT structures with 1x2 blocks. Thus, IF one
> was to crossref I would suggest to do it to A123764, but perhaps this is a
> little hard to explain in the limited space of an OEIS entry.
>
> But Watson has even more restrictions; he looks only at MAX HEIGHT
> buildings with one block at each level. This is exactly what LEGO did wrong
> when they claimed that A123828(6)=102981500 [the number they wanted was
> (46^5+2^5)/2, but they got the last digit wrong too].
>
> Of course counting FLAT and MAX HEIGHT structures with 2x2 or 1x2 blocks
> is easy, the formula is 3^(n-1) or (3^(n-1)+1)/2 depending on whether or not
> you want to identify up to symmetry. Watson knows this and tries to count
> how many of these structures would tip over if placed on a flat surface,
> essentially computing the x-coordinate of the center of gravity. So I think
> the kind of LEGO counting done by Watson is quite different from the kind of
> LEGO counting we do.
>
>     {Btw, A007576 is indeed the same as A086821}
>     {If somebody was interested I could probably quite easily compute how
> many of "our" LEGO buildings are stable in the sense of Watson}
>
> Regarding Zeilberger's LEGO counts, they all relate, as far as I recall
> from a relatively hard look some time ago, to buildings with blocks of
> varying size, and hence are quite different in nature from what we do. In
> that case, n is not a number of blocks, but the total volume/weight of the
> structure, so that's a very different ball game.
>
> Best regards,
> Soren
>
>
>
>
>
> On 01/11/06 18:58, "Jonathan Post" <jvospost3 at gmail.com> wrote:
>
> I like the new seqs about Lego, such as
> A123762 <http://www.research.att.com/%7Enjas/sequences/A123762><http://www.research.att.com/%7Enjas/sequences/A123762>
>   Number of ways, counted up to symmetry, to build a contiguous building
> with n *LEGO* blocks of size 1x2.
>
> Ironically timed, as *"Recent restructuring and production cuts have left
> Lego unable to fill orders
> <http://www.cbc.ca/money/story/2006/10/31/legoproduction.html><http://www.cbc.ca/money/story/2006/10/31/legoproduction.html>for the upcoming holiday season. Affected products include Duplo bricks,
> Lego City sets, and (horror of horrors!) Star Wars and Lego Technik sets."
> * According to the article Lego stands to lose $127 million in holiday
> sales.
>
> http://www.cbc.ca/money/story/2006/10/31/legoproduction.html
>
> It might be useful to add crossrefs to earlier seqs such as:
>
> A007576 <http://www.research.att.com/%7Enjas/sequences/A007576><http://www.research.att.com/%7Enjas/sequences/A007576>
>   Number of maximally stable towers of 2 X 2 *LEGO* blocks.
>
>
> A082679 <http://www.research.att.com/%7Enjas/sequences/A082679><http://www.research.att.com/%7Enjas/sequences/A082679>
>   Number of *Lego* towers, one piece per floor, where every floor is
> perpendicular to the one below it (so we have a kind of 3-dimensional zigzag
> pattern).
>  -- Jonathan Vos Post
> [wondering if my son's earlier intense exposure to Duplo and Lego led him
> towards his double B.S. in Math and Computer Science]
>
>
>
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