Packing boxes
Crammy
inches,
that you pack into a crate measuring 4 feet x 5 feet x 6 feet?
Crammy
David W. Cantrell
> What's the most number of boxes, each measuring 7 inches x 11 inches x 13
> inches, that you pack into a crate measuring 4 feet x 5 feet x 6 feet?
You can certainly pack 192 of the boxes in the crate. The arrangement is
fairly simple. Whether one can do better than that, I don't know.
David
Robert Israel
From the volume, 207 is an upper bound.
I can get 194 with:
2 x 4 x 10 array of boxes in the 13 x 11 x 7 orientation
2 x 4 x 10 array of boxes in the 11 x 13 x 7 orientation
2 x 2 x 6 array of boxes in the 13 x 7 x 11 orientation
2 x 1 x 5 array of boxes in the 11 x 7 x 13 orientation
It's probably not optimal.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
David W. Cantrell
> In article <d96jra$p2a$1...@nwrdmz01.dmz.ncs.ea.ibs-infra.bt.com>,
> Crammy <cra...@hotmail.com> wrote:
> >What's the most number of boxes, each measuring 7 inches x 11 inches x
> >13 inches,
> >that you pack into a crate measuring 4 feet x 5 feet x 6 feet?
>
> From the volume, 207 is an upper bound.
>
> I can get 194 with:
>
> 2 x 4 x 10 array of boxes in the 13 x 11 x 7 orientation
> 2 x 4 x 10 array of boxes in the 11 x 13 x 7 orientation
> 2 x 2 x 6 array of boxes in the 13 x 7 x 11 orientation
> 2 x 1 x 5 array of boxes in the 11 x 7 x 13 orientation
>
> It's probably not optimal.
My earlier packing, getting 192 boxes in the crate, was just a quick idea.
I can now get 203 boxes in the crate. And of course that might not be
optimal either.
David
David W. Cantrell
Since nobody claimed a better packing yet, I supposed I should describe
mine. But in the process of writing the description, I've come up with an
even better packing, which gets 205 boxes in the crate.
Suppose that we place one of the crate's faces which is 6' by 4' on the
floor. Peering down into the crate, my bottom layer of 24 boxes, 7" high,
would look something like the following.
[Please view in a fixed-width font.]
_____________________________
| 13 | 13 | 11 | 11 |
|11 | | | |
|_______|_______| | 13|
| | |______|______|
|11 | | | |
|_______|_______| | |
| | | | 13|
|11 | |______|______|
|_______|_______| | |
| | | | | |
|13 | |X| | 13|
| | |_|______|______|
|______|______| | |
| | | | 11|
|13 | |_______|_______|
| | | | |
|______|______| | 11|
| | |_______|_______|
|13 | | | |
| | | | 11|
|______|______|_______|_______|
11 11 13 13
The space in the middle, marked X, is unusable, being only 6" by 4".
We then put in the crate 6 more layers of 24 boxes, exactly like the
bottom layer. So we have, thus far, packed 7 layers of 24 boxes.
Conveniently, above the top layer, we have an 11" space remaining. It
may be packed to look something like
__________________________
| 7 | 7 | 7 | 7 | 7 | 13 |
| | | | | |______|7
|13 | | | | | |
|___|___|___|___|___|______|7
| | | | | | |
| | | | | |______|7
|13 | | | | | |
|___|___|___|___|___|______|7
| | | | | | |
| | | | | |______|7
|13 | | | | | |
|___|___|___|___|___|______|7
| | | | | | |
| | | | | |______|7
|13 | | | | | |
|___|___|___|___|___|______|7
| | | | | | |
| | | | | |______|7
|13 | | | | | |
|___|___|___|___|___|__ X |
|7 | | | |
|_______|_______|______|___|
13 13 13
Again, the space marked X is unusable (although it might not seem so, due
to the crude nature of my ASCII drawing).
The top layer now has 37 boxes, and each of the 7 lower layers has 24
boxes. Altogether then, we have packed 205 boxes.
Of course, this packing still might not be optimal.
David W. Cantrell
Gerald Oliver Swift
"David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
news:20050623124606.796$W...@newsreader.com...
David
205 boxes was the answer I was given many, many years ago when I was an
undergraduate student.
It seems highly improbable, even now, (given the dimensions of the boxes,
the dimensions of the case & the fact that 207 is the upper bound) that 205
will actually pack.
I am so, so pleased that after all these years someone has provided me with
a solution to the problem.
Many, many thanks
Crammy