I would think a cube.  For a given volume-weight, divide by 3 and make a cube 
of that dimension on each side.



On Monday 11 June 2007 06:08, Russell McMahon wrote:
> A diversion for those who can be bothered:
>
> Kristin is studying in Australia.
> The weather is getting cold lately.
> Russell & Valerie want to send her a box full of clothes, eiderdown,
> text books and more.
> The largest box that can be sent by parcel post is
>
>     Largest side <= 1000 mm
>     Sum of L +     2 x (D + H)  <= 2000 mm = Volume.weight = VW
>
>         L     length
>         D    depth
>         H    Height
>
> True volume is of course L x D x H
>
> Assignment for students:
>
>
>                 What box dimensions maximise the true volume that can
> be sent?
>                 Assume a rectangular box.
>
>                 Report time to solve.
>
>
> The answer seems somehow obvious in retrospect but was annoyingly
> unintuitive (to me).
>
> Clue - of sorts:        John Crook will tell me that the answer is
> obvious on inspection using information theory (he'll be right).
>
>
>
>
>             Russell
>
>
>
>
> _____________________________
>
> Notes:
>
> 1.    I sent the parcel this evening - all 17.7 kg of it.
>
> 2.    Volume.Weight is actually a mongrel measure that is neither
> volume per weight.
>
> 3.    2 x (D+H) is the "girth" - the distance around the box.
> They seem to do it this way because they can do it with a pencil and
> paper (or in their head) more easily than using multiplication.
> What they REALLY want is volume.
> The ideal shape (from a sender's point of view) maximises the true
> volume per volume.weight.
>
> ie              maximises     L x D x H / ( L + 2x(D+H))
>
> However, for a fixed volume.weight, maximising volume does just as
> well so maximising
>
>     L x D x H while keeping
>
>         L + 2 x (D+H) = 2000 mm does the same thing.
>
> 4.    We had a nice black plastic box with lid, plastic clip in
> rollers and dimensions that *just* exceeded the allowable maximum VW
> if you took the very outer measurements in each case. As this was
> handles etc, and as the cross section was a trapezoid that was
> substantially smaller than D x H so true VW was well under the limit,
> it clearly met the spirit of the rules and MAY have met the letter
> depending on the mood of the clerk concerned. I used it. They didn't
> bother measuring it.
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