Where did all these dowels come from?

[Prof. Templeton]

This may have been asked before since it was taken straight from bn’s playbook.

You have an unlimited supply of dowels all of the same length and they can only be connected at their ends. A triangle made by joining three dowels will be a rigid form, but a square made by connecting four dowels can be easily distorted into other shapes without breaking or bending a dowel or disconnecting the ends. As if the end connections were hinged. In three dimensional space the simplest way to brace the square would be to add eight more dowels the form an octahedron.

Let’s assume, however that we are limited to only two dimensions. Without bending, breaking or overlapping dowels, only connecting them at the ends and remaining on a two dimensional plane we can still brace the square by attaching triangles to it. Here is an example with 35 total dowels.

The question is: What is the fewest number of dowels required to hold a square firm?

[Faizaan]

I am sitting here with my homework tossed aside constantly rearranging a large number of tooth-picks in a desperate attempt to figure this out.

[Faizaan]

ooh! ooh! WAIT!

27 dowels

Arranged as such

That was terribly crud(e).

[Prof. Templeton]

ooh! ooh! WAIT!

27 dowels

Arranged as such

That was terribly crud(e).

I don't think that will hold, but applaud the effort.

In this picture I rotated your triangle bracing on the sides so that they held their original shape. I then deformed the floppy squares in the center.

[curr3nt]

gah nevermind...

Edited October 4, 2012 by curr3nt

[r8erred]

16

[curr3nt]

-edit-

*sigh* nope

Can I guess that 35 is the minimum?

Edited October 4, 2012 by curr3nt

[curr3nt]

I count 34!

To alter the square one pair of angles has to get bigger while their adjacent angles get smaller.

Two of the triangle arms prevent the angle between them from getting bigger but the angle can get smaller. By using three arms this prevents the angles adjacent to each other in the square from getting bigger. Therefore the square can't alter shape.

Edited October 4, 2012 by curr3nt

[plainglazed]

[curr3nt]

change the square's angle between the two triangles from a 90 into a 60. That would make the edges of the two triangles parallel and close the gap between the triangles.

Edited October 4, 2012 by curr3nt

[Faizaan]

I think...

[Faizaan]

Nevermind. That was terrible. I left the 'derp-mode' switch on.

I don't see how that could move.

[curr3nt]

Recreate it with a rhombus with angles of 60 and 120 instead of squares for an idea of how it can change.

Top square and bottom square are flattened and middle square's left side drops down.

I already tried that. Made some pretty patterns when I kept expanding it.

[Rob_Gandy]

7?

Square.bmp

Seems to work with pencils.

[phaze]

How thick and long is the dowel? Given the right proportions Rob may be a winner.

Note the triangle is not connected to the square as end needs to be connected to end but the triangle does not need to be connected to the square?

Note while if we assume infinite thinness of the dowel the triangle has to be larger than the square for this to work if the dowel has the right thickness this can be accounted for.

[curr3nt]

Since we are dealing with a 2-D world, I was going on the assumption that the dowels were lines.

[Prof. Templeton]

Dowels would be lines for our purposes and they can only be connected at the ends.

[Faizaan]

Since we are dealing with a 2-D world, I was going on the assumption that the dowels were lines.

A 3-dimensional object viewed on a 2-dimensional plane does not become 1-dimensional.

[bonanova]

I won't post. I looked in my play book.

Actually I have a couple of them.

PT nice to see you again.

[plainglazed]

[curr3nt]

So... is it possible to get less than 34?

[Prof. Templeton]

So... is it possible to get less than 34?

Yes. pg may be on the road to a more efficient solution with a small modifications to his current design. I'm glad to see he didn't make the common mistake of trying to join the connections (circled) on the bottom and eliminating a number of supports on the top.

[plainglazed]

[bonanova]

OP says "Without bending, breaking or overlapping dowels, only connecting them at the ends ..."

[CaptainEd]

I see that other solutions have had the triangles impinge on the square, without connecting to it.

My question about the 33 solution is that I imagine the top and bottom squares collapsing to the left and right respectively, while still remaining pinched between the crab claws. So I think the squares are not actually firmed up.

[Yoruichi-san]

I've been trying to avoid thinking about this since I can foresee not getting any work done for the next week...*whistling*

Since Bon-chan has indicated he already knows the answer, perhaps it should be taken as a hint to the solution...;P