Elegant Problem. Elegant Solution?
Robert DeSoucey
You're walking with a ladder down a hallway with a width "a". You
have to turn a corner into a hallway with width "b".
What is the longest ladder you can turn around this corner? Of
course the ladder is a line with no width dimension.
The more I play with it, the more I enjoy it. It even comes close
to the three body problem if you play it right.
Bob
Mario O. Bourgoin
Robert DeSoucey (boc...@ix.netcom.com) writes:
You're walking with a ladder down a hallway with a width "a". You
have to turn a corner into a hallway with width "b".
What is the longest ladder you can turn around this corner? Of
course the ladder is a line with no width dimension.
What is the height of the ceiling?
--Mario
Per Erik Manne
> >I've been flirting with this one for 30 years. Any takers?
> >have to turn a corner into a hallway with width "b".
> > What is the longest ladder you can turn around this corner? Of
> >course the ladder is a line with no width dimension.
> >
> >to the three body problem if you play it right.
> >Bob
Now the really nice version of this is to ask what is the largest
sofa you can turn around the corner. (The sofa is a rigid
2-dimensional object, and you want to get it around the corner
in the hallway without tilting it. What is the largest possible
area the sofa can have?)
--
Bergen,
Per Manne
p...@hamilton.nhh.no
Allen Windhorn
Jos van Kan <j.va...@math.tudelft.nl> writes:
>>boc...@ix.netcom.com (Robert DeSoucey) wrote:
>>>I've been flirting with this one for 30 years. Any takers?
>>> You're walking with a ladder down a hallway with a width "a". You
>>>have to turn a corner into a hallway with width "b".
>>> What is the longest ladder you can turn around this corner? Of
>>>course the ladder is a line with no width dimension.
> L = (a^(2/3) + b^(2/3))^(3/2)
>
There was a related problem I read about what the largest sofa was which
could be taken around a corner, and what the optimum shape was for it.
May have been one of Ian Stewart's books. Anyone remember?
------------
Allen Windhorn
Kato Engineering N. Mankato, MN 56002
Kerry M. Soileau
did not specify how high the ceiling is, so let's assume it's
infinite. Place one end of your ladder on any point in the
intersection of the two hallways. Swing the other end of the ladder
upwards through the first hallway's "airspace" until it is vertical.
Then swing it down through the second hallway's "airspace" until it
is again horizontal. This procedure's success is independent of the
ladder's length.
Regards,
Kerry
Benjamin J. Tilly
Per Erik Manne <p...@hamilton.nhh.no> writes:
> > boc...@ix.netcom.com (Robert DeSoucey) wrote:
> > >I've been flirting with this one for 30 years. Any takers?
> > > You're walking with a ladder down a hallway with a width "a". You
> > >have to turn a corner into a hallway with width "b".
> > > What is the longest ladder you can turn around this corner? Of
> > >course the ladder is a line with no width dimension.
> > >
> > > The more I play with it, the more I enjoy it. It even comes close
> > >to the three body problem if you play it right.
> > >Bob
>
> Now the really nice version of this is to ask what is the largest
> sofa you can turn around the corner. (The sofa is a rigid
> 2-dimensional object, and you want to get it around the corner
> in the hallway without tilting it. What is the largest possible
> area the sofa can have?)
Must my sofa be rectangular? :-)
Ben Tilly
Chris Thompson
writes:
|>
|> Now the really nice version of this is to ask what is the largest
|> sofa you can turn around the corner. (The sofa is a rigid
|> 2-dimensional object, and you want to get it around the corner
|> in the hallway without tilting it. What is the largest possible
|> area the sofa can have?)
and in article <47oiii$e...@Oak.IC.Mankato.MN.US>, wind...@ic.mankato.mn.us
Allen Windhorn) writes:
|>
|> There was a related problem I read about what the largest sofa was which
|> could be taken around a corner, and what the optimum shape was for it.
|> May have been one of Ian Stewart's books. Anyone remember?
It is often a piano rather than a sofa, which does have a better sense of
rigidity to my mind.
Try section G5 in [UPIG] for references. The Shephard piano is a good try:
E_______F
. | | .
. | , . | .
. |. .| .
A----B C----D
(ObApology for ASCII art) - a rectangle BCFE with quarter circles ABE, CDF
added and a semicircular bite with diameter BC taken out. AB = CD = BE = CF
= 1, the width of the corridors. It's fairly easy to see that this goes round
the corner: think "right angle in a semicircle". The area is maximised if
BC = EF = 4/pi, at 2/pi + pi/2 = 2.2074... The best known solution (as of
the 1st edition of [UPIG], anyway) is 2.2156...
There is also the "Conway car": what is the maximum area rigid 2-D shape that
can reverse in a T-junction, all roads having unit width?
[UPIG] Unsolved Problem in Geometry
H.T. Croft, K.J. Falconer, R.K. Guy
Springer (1991 for 1st edition, but there is a 2nd out)
Chris Thompson
Email: ce...@cam.ac.uk
AS Lim
>In <DHo7C...@dutiws.twi.tudelft.nl>,
> Jos van Kan <j.va...@math.tudelft.nl> writes:
>>>boc...@ix.netcom.com (Robert DeSoucey) wrote:
>>>>I've been flirting with this one for 30 years. Any takers?
>>>> You're walking with a ladder down a hallway with a width "a". You
>>>>have to turn a corner into a hallway with width "b".
>>>> What is the longest ladder you can turn around this corner? Of
>>>>course the ladder is a line with no width dimension.
>> (nice math deleted)
>> L = (a^(2/3) + b^(2/3))^(3/2)
>>
>There was a related problem I read about what the largest sofa was which
>could be taken around a corner, and what the optimum shape was for it.
>May have been one of Ian Stewart's books. Anyone remember?
>------------
>Allen Windhorn
>Kato Engineering N. Mankato, MN 56002
If the height of hall is h and the ladder can be carried any way
suitable the new ladder's length will be equal to L^2 +h^2 where
L is the length of ladder if carried horizontally. And this new
ladder is equal to [(a^2/3 +b^2/3)^3] + h^2
Fred W. Helenius
writes:
>Benjamin J. Tilly wrote:
>> Per Erik Manne <p...@hamilton.nhh.no> writes:
[snip]
>> > Now the really nice version of this is to ask what is the largest
>> > sofa you can turn around the corner. (The sofa is a rigid
>> > 2-dimensional object, and you want to get it around the corner
>> > in the hallway without tilting it. What is the largest possible
>> > area the sofa can have?)
>>
>> Must my sofa be rectangular? :-)
>>
>> Ben Tilly
>
>Then it wouldn't be too difficult, would it? Let's say the sofa
>is bounded by finitely many smooth curves.
>
>Note: I assume that the problem is unsolved. Does anyone know
>any good upper bounds on the area?
I have heard that the problem was solved a couple of years ago by
Joseph Gerver, but I don't have a reference. His solution involved
trimming a few bits off the shape called the "Shephard Piano" (after
Geoffrey Shephard) so the whole shape could be expanded a bit.
Shephard's work is described in Croft & Guy's "Unsolved Problems
in Geometry"; Gerver's work may have made it into the new edition,
which I haven't seen.
--
Fred W. Helenius <fr...@ix.netcom.com>