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        riddles >> hard >> Stabilise the square!
        
(Message started by: Wonderer on Dec 17th, 2005, 12:47am)
Title: Stabilise the square!
Post by Wonderer on Dec 17th, 2005, 12:47am
Stabilise the square!

You are given 4 identical chopsticks; each has a small connecter at both ends.  With the connectors, you can use the 4 chopsticks to form a perfect square.  However, this square is not stable.  Applying pressure will force the square to change shape.  

Now, you are given an infinite amount of such chopstick.  Can you find a way so that this square can be stabilised?

Please note, you can only connect chopsticks with their ends.  
Title: Re: Stabilise the square!
Post by JocK on Dec 17th, 2005, 1:18am
With four more chopsticks I'd build a [hide]pyramid with a square base[/hide].
Title: Re: Stabilise the square!
Post by Wonderer on Dec 17th, 2005, 1:36am

on 12/17/05 at 01:18:08, JocK wrote:
With four more chopsticks I'd build a [hide]pyramid with a square base[/hide].


No, I don't think so.  This was also the first answer I came across,  bust later realized that the square can still change shape into a 3D object.  Even building a Octhedron structure could not stabilize the square.

The square has to be stabilized in 2D.
Title: Re: Stabilise the square!
Post by JocK on Dec 17th, 2005, 2:27am

on 12/17/05 at 01:36:17, Wonderer wrote:
No, I don't think so.  This was also the first answer I came across,  bust later realized that the square can still change shape into a 3D object.  Even building a Octhedron structure could not stabilize the square.

The square has to be stabilized in 2D.


Are you saying you can build an octahedron with equal edge lengths that is not regular?  ???
Title: Re: Stabilise the square!
Post by Wonderer on Dec 17th, 2005, 3:11am

on 12/17/05 at 02:27:15, JocK wrote:
Are you saying you can build an octahedron with equal edge lengths that is not regular?  ???



Oooops, sorry. My mistake.  What was I thinking .  Octahedron does work.

Now, what if you are not allowed to build 3D structures?  You can only connect chopsticks in 2D.  Can you still stabilize the angles in the square?
Title: Re: Stabilise the square!
Post by Icarus on Dec 17th, 2005, 7:16am
The only rigid structure that you can make is a triangle. Further, since any side longer than 1 that you build will have a hinge in the middle, the only rigid triangle that you can build is an equilateral triangle of sidelength one.

Can you build a square from equilateral triangles? No. All angles will be multiples of 60o. So right angles cannot be constructed.

Now, I have assumed that your chopsticks can only meet at their ends, since crossing elsewhere would require 3-D to allow one to go over the other. If you accept this minor departure from 2-D, however, you get more freedom, and can actually build 90o angles. I do not believe you can make them rigid, but this is harder to see.
Title: Re: Stabilise the square!
Post by Sjoerd Job Postmus on Dec 17th, 2005, 8:24am
The only rigid structure you can make without having surrounding structures is a triangle, as Icarus said.

But, you can expand on this.

You can also have two linked diamonds stable.

How to build two stable diamonds?

Start with a regular hexagon with radius one chopstick.

remove one chopstick, notice it's still stable? [in 2d].

Now, if you put two diamonds in this place, with angle 30*... you notice these diamonds aren't stable, but they will be when you put a triangle on the  open part of the diamonds.

Now...

place another diamond in the other direction, meeting back at the hexagon. Notice a 90* angle?

Now, we don't have a stable square yet, but we're getting there.

Just add two more triangles to your structure, and a square.

At least, I think it is stable... can anyone doublecheck?
Title: Re: Stabilise the square!
Post by Joe Fendel on Dec 17th, 2005, 11:15am
I like your creativity, Sjoerd, but it doesn't look stable to me.  I could be wrong, though.

It looks to me like the lower-left two-triangle diamond can, for example, be "pushed" up against the almost-hexagon.  This has the effect of pushing the upper-left triangle up and to the right and also deforming the square into a rhombus.

But hey, I don't have any better ideas!
Title: Re: Stabilise the square!
Post by towr on Dec 17th, 2005, 3:11pm
What if you surround a hexagon with 6 squares (connected to the hexagon) and 6 triangles (connecting the squares)?
Title: Re: Stabilise the square!
Post by JohanC on Dec 17th, 2005, 3:12pm
What about the following?
[hide]
1) Create a chain of equilateral triangles to get a stable straight line of 3 chopsticks
2) do the same for 4 chopsticks
3) and once again for 5 chopsticks
4) form a triangle with sides 3,4 and 5, making sure the construction chopsticks are all outside the triangle; if memory serves well, this should form a right angled triangle;
5) add two more chopsticks in the right corner of the triangle to form the square[/hide]
Title: Re: Stabilise the square!
Post by Wonderer on Dec 17th, 2005, 3:14pm

on 12/17/05 at 15:12:26, JohanC wrote:
What about the following?
[hide]
1) Create a chain of equilateral triangles to get a stable straight line of 3 chopsticks
2) do the same for 4 chopsticks
3) and once again for 5 chopsticks
4) form a triangle with sides 3,4 and 5, making sure the construction chopsticks are all outside the triangle; if memory serves well, this should form a right angled triangle;
5) add two more chopsticks in the right corner of the triangle to form the square[/hide]


Bingo!!  Correct answer!!! :D
Title: Re: Stabilise the square!
Post by towr on Dec 17th, 2005, 3:16pm
That obviously works, very good
So the next question is, do you really need to use that many chopsticks?
Title: Re: Stabilise the square!
Post by JohanC on Dec 17th, 2005, 3:42pm

on 12/17/05 at 15:16:33, towr wrote:
That obviously works, very good

Thanks.
I'm attaching a drawing.


on 12/17/05 at 15:16:33, towr wrote:
So the next question is, do you really need to use that many chopsticks?

Probably not.
Still some food for thought, while waiting for the chopstick food to be served ....
Title: Re: Stabilise the square!
Post by Barukh on Dec 17th, 2005, 11:32pm
Nicely done, Johan!

BTW, are the chopsticks allowed to cross?
Title: Re: Stabilise the square!
Post by towr on Dec 18th, 2005, 7:37am
Here's a drawing for the solution I offered a minute before JohanC gave his (inspired by Sjoerd's solution).
It needs fewer chopsticks, and also has more squares.
And it has a certain elegance imo.

Of course you can still remove at least one more chopstick, from the hex.
Title: Re: Stabilise the square!
Post by SMQ on Dec 18th, 2005, 8:49am

on 12/18/05 at 07:37:55, towr wrote:
Of course you can still remove at least one more chopstick, from the hex.

In fact, can't you remove all six inner chopsticks from the hex?  displacing any of the corners of the hex requires distorting the equilateral triangle outsie that corner, yes?

--SMQ
Title: Re: Stabilise the square!
Post by towr on Dec 18th, 2005, 8:57am

on 12/18/05 at 08:49:07, SMQ wrote:
In fact, can't you remove all six inner chopsticks from the hex?  displacing any of the corners of the hex requires distorting the equilateral triangle outsie that corner, yes?

No, if you remove the six inner sticks, you get 4 parallel chopsticks in a row, with nothign to stop them tilting. And that three times in different directions.

Maybe two, on the outside the hex, one on one side and the other opposite.
Title: Re: Stabilise the square!
Post by Joe Fendel on Dec 18th, 2005, 9:26am

on 12/18/05 at 07:37:55, towr wrote:
Here's a drawing for the solution I offered a minute before JohanC gave his (inspired by Sjoerd's solution).
It needs fewer chopsticks, and also has more squares.
And it has a certain elegance imo.

Of course you can still remove at least one more chopstick, from the hex.


This also looks unstable to me.  It seems like you could rotate all 6 outer triangles simultaneously so that the shape is a star-of-david, with 12 chopsticks doubled, for example.
Title: Re: Stabilise the square!
Post by towr on Dec 18th, 2005, 9:38am
Damn.. you're right.. :-[

Could you stabilize it by overlapping two or three of these? (Where the overlap between two is a square with a triangle on both sides)
Title: Re: Stabilise the square!
Post by towr on Dec 18th, 2005, 9:49am
Two are probably enough, but certainly three should be stable, right?

I should get out my Lego, I think :P
Title: Re: Stabilise the square!
Post by towr on Dec 18th, 2005, 10:29am
meh.. doesn't work either :(
I suppose I should just give up this approach..
Title: Re: Stabilise the square!
Post by Sjoerd Job Postmus on Dec 18th, 2005, 11:06am

on 12/18/05 at 10:29:36, towr wrote:
meh.. doesn't work either :(
I suppose I should just give up this approach..

It might not work, but it certainly looks cool...

Seems like the 3-4-5 is the only certainty now...
Title: Re: Stabilise the square!
Post by towr on Dec 18th, 2005, 11:13am

on 12/18/05 at 11:06:40, Sjoerd Job Postmus wrote:
Seems like the 3-4-5 is the only certainty now...
And any other Pythagorean triple (e.g. 5 12 13), but those are all much larger.
Title: Re: Stabilise the square!
Post by Barukh on Dec 18th, 2005, 11:31am
There exist configurations with fewer chopsticks. How fewer - depends on whether it is allowed or not to cross them.
Title: Re: Stabilise the square!
Post by Joe Fendel on Dec 19th, 2005, 12:21pm
Well, if they can cross, I think I can do it with 33 sticks.
Title: Re: Stabilise the square!
Post by towr on Dec 19th, 2005, 2:40pm
I hope you don't mind I trimmed some of the white off that image, it was a bit large.
Title: Re: Stabilise the square!
Post by Joe Fendel on Dec 19th, 2005, 2:46pm
Not at all.  I don't know how to make a jpeg except by making a powerpoint and saving it as a jpeg.  Like the farmer said to the raincloud, thanks for the crop assistance.
Title: Re: Stabilise the square!
Post by Joe Fendel on Dec 19th, 2005, 2:56pm
And I think I can see how to do it with 35 non-crossing, but I'm scared to make another jpeg...    :-[
Title: Re: Stabilise the square!
Post by towr on Dec 19th, 2005, 3:37pm
You could get irfanview, it's a free image viewer with some basic manipulation. You can just open the file, select a rectangular area, and then choose 'crop' from the edit menu, and save it.
Or otherwise, just post it, and I'll fix it in the morning ;)
Title: Re: Stabilise the square!
Post by Icarus on Dec 19th, 2005, 7:24pm
If you have Windows, you can also use Paint (under the accessories menu). It allows you to crop images and save them in a number of formats.
Title: Re: Stabilise the square!
Post by towr on Dec 20th, 2005, 12:28am
Actually Paint doesn't necessarily allow you to safe in a number of formats. There's some chance it'll just be a bmp, but with a different extension if you're not carefull.
Title: Re: Stabilise the square!
Post by Barukh on Dec 20th, 2005, 4:00am
[quote author=Joe Fendel link=board=riddles_hard;num=1134809282;start=25#27 date=12/19/05 at 14:56:27]And I think I can see how to do it with 35 non-crossing, but I'm scared to make another jpeg...    :-[/quote]
It can be done with further sticks.  ;)
Title: Re: Stabilise the square!
Post by Joe Fendel on Dec 20th, 2005, 5:32am

on 12/20/05 at 04:00:32, Barukh wrote:
further sticks

"Further"? As in fewer?
Title: Re: Stabilise the square!
Post by Barukh on Dec 20th, 2005, 7:58am

on 12/20/05 at 05:32:18, Joe Fendel wrote:
"Further"? As in fewer?

Yes, of course!  ;D
Title: Re: Stabilise the square!
Post by Grimbal on Dec 21st, 2005, 3:43pm
29 and no crossings  8)
Title: Re: Stabilise the square!
Post by towr on Dec 21st, 2005, 4:02pm
very nice.
Title: Re: Stabilise the square!
Post by Barukh on Dec 22nd, 2005, 4:04am

on 12/21/05 at 15:43:43, Grimbal wrote:
29 and no crossings  8)

Indeed, very nice, but can be improved further.

By the way, how do you show that a configuration is stable?
Title: Re: Stabilise the square!
Post by towr on Dec 22nd, 2005, 4:59am

on 12/22/05 at 04:04:24, Barukh wrote:
By the way, how do you show that a configuration is stable?
That can be difficult. But as long as you're just working with triangles, each triangle is stable if the sides are stable. And a square is stable if two opposite points are stable.
Title: Re: Stabilise the square!
Post by Grimbal on Dec 22nd, 2005, 5:19am

on 12/22/05 at 04:04:24, Barukh wrote:
Indeed, very nice, but can be improved further.

By the way, how do you show that a configuration is stable?

It can be improved without crossing?  With crossings I can bring it down to 21 in an obvious way.

Stabillity.
There are 29 sticks, each stick has 3 degrees of freedom.  Each connection removes 2 degrees of freedom. There are 42 connections (k sticks joining counts as k-1 connections).  That makes 29*3-42*2 = 87 - 84 = 3 degrees of freedom.  That is just enough to move the whole figure.
This assumes there is no redundant stick.  That is the difficult point.

A better way to show it is:
- one stick is a rigid construct.
- 3 rigid constructs that are connected 2 by 2 make a rigid construct if the connection points are not aligned (and therefore distinct).
- my figure can be built that way.
Title: Re: Stabilise the square!
Post by Barukh on Dec 22nd, 2005, 5:55am

on 12/22/05 at 05:19:47, Grimbal wrote:
It can be improved without crossing?

Yes.
Title: Re: Stabilise the square!
Post by JocK on Dec 22nd, 2005, 10:36am

on 12/22/05 at 05:19:47, Grimbal wrote:
It can be improved without crossing?  With crossings I can bring it down to 21 in an obvious way.


With crossing you should be able to bring it down below 21.
Title: Re: Stabilise the square!
Post by Wonderer on Dec 22nd, 2005, 9:55pm

on 12/21/05 at 15:43:43, Grimbal wrote:
29 and no crossings  8)


Very nice!  
Title: Re: Stabilise the square!
Post by Wonderer on Dec 22nd, 2005, 9:55pm

on 12/22/05 at 05:55:27, Barukh wrote:
Yes.


Really????! how?
Title: Re: Stabilise the square!
Post by Barukh on Dec 23rd, 2005, 12:36am

on 12/22/05 at 21:55:54, Wonderer wrote:
Really????! how?

See the following page (http://mathworld.wolfram.com/BracedPolygon.html) and a link given in the references which has interesting results on higher polygons.

The solution shown there passes "Grimbal's test" of rigidity, although frankly, I don't understand it fully.
Title: Re: Stabilise the square!
Post by Wonderer on Dec 23rd, 2005, 3:07am

on 12/23/05 at 00:36:52, Barukh wrote:
See the following page (http://mathworld.wolfram.com/BracedPolygon.html) and a link given in the references which has interesting results on higher polygons.

The solution shown there passes "Grimbal's test" of rigidity, although frankly, I don't understand it fully.


unbelievable... :o
Title: Re: Stabilise the square!
Post by SMQ on Dec 23rd, 2005, 6:07am

on 12/23/05 at 00:36:52, Barukh wrote:
although frankly, I don't understand it fully.

The solution, or Grimbal's test?  Grimbal's test basically says a triangle formed of rigid elements is itself a rigid -- no mystery there, I hope.

For the non-crossing solution, the bottom section extends two sides of the square to form two horizontal phantom lines (dashed) of length sqrt(2), then the top section used two sqrt(1),sqrt(2),sqrt(3) right triangles to stabilize the figure.  Because of the symmetry the vertical center link must be vertical, and because it's of length 1 the square must be a perfect square.

The crossing solution uses the fact that a right triangle with legs 1 + sqrt(2) and 1 - sqrt(2) has a hypotenuse of sqrt(3), and uses two crossed "diamonds" to stabilize the figure.

Hope that helps. :)

--SMQ
Title: Re: Stabilise the square!
Post by Sjoerd Job Postmus on Jan 10th, 2006, 5:29am
I found yet another solution, one that assumes that sticks MAY NOT cross, ever... Not even while de-stabalizing.

The bottom line is stable, duh.

Look at the segment left to the square, it can only fall over to the right. But, because the link, the other segment must too, but it can only fall over to the left. So, you can't move either of them in a direction. So, it's stable.


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