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        riddles >> easy >> Elastic Bands
        
(Message started by: James Fingas on Oct 24^th, 2003, 7:11am)

Title: Elastic Bands
Post by James Fingas on Oct 24^th, 2003, 7:11am

You have two elastic bands of the same size, and an unbound document. You want to put the elastic bands around the document. If you loop each band once around (two loops total), the elastic bands are too loose to hold the document together. If you loop each band around twice (four loops total), the elastic bands are too tight.

Is it possible to loop the elastic bands around a total of 3 times (3/2 of the way around each)?

Title: Re: Elastic Bands
Post by TenaliRaman on Oct 24^th, 2003, 8:26pm

I was about to say 2 loops of one and a loop of the other before i saw "3/2 of the way around each".

so i say,
::[hide]cut the two elastic bands and fuse the two bands at their end points and loop this big band around the document thrice.[/hide]::

Title: Re: Elastic Bands
Post by THUDandBLUNDER on Oct 24^th, 2003, 8:50pm

Quote:

fuse the two bands at their end points

Perhaps with a blowtorch? Why didn't I think of that? ::)

Title: Re: Elastic Bands
Post by TimMann on Oct 24^th, 2003, 9:49pm

I think James meant to say that the bands are not to be cut, glued, or knotted. I found a solution under those constraints pretty quickly by playing with a couple of real rubber bands.

It's too soon for spoilers, so I'll make a link to my ugly drawing of the solution instead of attaching it. See http://tim-mann.org/image/bands.jpg.

Title: Re: Elastic Bands
Post by TenaliRaman on Oct 24^th, 2003, 11:19pm

T&B,
one could always staple the ends if the bands are broad but blow torches work most of the times ;D

BTW, since no conditions were explicitly mentioned i uphold my answer to be true :P

Anyways i had found two configurations which could satisfy the result(under the assumed conditions).One of which is same as that of TimMann's and the other is something like this,(be prepared for even uglier drawing

Title: Re: Elastic Bands
Post by TimMann on Oct 24^th, 2003, 11:40pm

The alternative solution in TenaliRaman's drawing stretches one of the bands twice as tight as the other. I.e., it's not the case that each band goes around 3/2 times -- the red one goes around twice and the black one goes around once. Unless I'm misunderstanding the drawing somehow.

Title: Re: Elastic Bands
Post by TenaliRaman on Oct 25^th, 2003, 12:03am

Consider the positions at equilibrium..

Note that the red band is not twisted.So it is bound to pull itself back and at equilibrium the loops should be around 3/2 each.

Title: Re: Elastic Bands
Post by TimMann on Oct 25^th, 2003, 12:35am

That makes no sense to me. Where is the document supposed to be in your drawing? If the document is horizontal and all three big vertical loops go around it, then the two that are part of the red band each go around all the way, and so the red band goes around twice, not 3/2 times. The black band goes around once (minus an insignificant part where it wraps around the red band).

Your explanation does apply to my drawing, but I already understand why it's correct. I was asking why you think yours is. ::)

Title: Re: Elastic Bands
Post by TenaliRaman on Oct 25^th, 2003, 4:47am

My drawing ability on my comp is very limited so if the following diagram is totally hidious, you can always call it "modern art".

[edit] ok "stretches" in the diag be replaced "constricts"[/edit]

Title: Re: Elastic Bands
Post by TimMann on Oct 25^th, 2003, 10:58am

Oh, I see. Then that's the same as the solution in my drawing. Your red band corresponds to my blue band, and your black band to my red band.

Title: Re: Elastic Bands
Post by Rezyk on Oct 25^th, 2003, 1:40pm

Here's another way:

Title: Re: Elastic Bands
Post by TenaliRaman on Oct 25^th, 2003, 9:02pm

You're right Tim!! :o
i never realised that !! ( say what we both could take some MSPaint class ;D )

Btw, nice diag Rezyk! and configuration seems proper as well and it seems to hold the document pretty tight !

Title: Re: Elastic Bands
Post by TimMann on Oct 25^th, 2003, 9:18pm

Rezyk's way isn't as different from ours as it perhaps looks. First note that the drawing is a bit off -- the blue band is stretched much more than the red one. If you correct the distortion by making the triple part of the blue band shorter and the red band corrspondingly longer, it looks a lot like my solution with red and blue swapped. The difference is the way the end of the blue band is wrapped around the red one instead of simply hooked through it.