Cut A Hole! In A Sheet Of Paper Problem!...

Dross · 2007-11-07 04:09:55

I think quiet appreciation is enough.

JohnnyReinB · 2007-11-07 21:36:08

ok, Then lets quietly celebrate?

Kargoneth · 2007-12-11 17:11:46

John E. Franklin wrote:

I am dumb; can you have a 1-dimensional rectangle of paper for the frame with zero area?

An interesting question . A rectangle is a two-dimensional object though

I believe the quote below is very similar to what you are visualizing (just using molecules as the frame).

JohnnyReinB wrote:

I think, for any sized paper, the largest hole you can cut (without doing the tricks of turning it into a hole with a larger area than the original) is one with one with an edges thickness of an atom/molecule's diameter. The percentage differs from paper to paper, so I don't know the percentage...if you destroy the atoms/molecules when cutting the paper, the paper would break down, and it would no longer be paper with a hole in it! It would just be a bunch of subatomic particles.

Agreed. It could also be worded that the instant you destroy one of the molecules making up the perimiter of said paper, the paper would become a string/sheet 1 atom wide, without a hole (since it has essentially gone from a ring to a rod, lol)

Anthony.R.Brown wrote:

Paper! as you know it!.. is not the Paper Known in the Enormous! World of Infinity!...

*Scratches head* Um... exclamation marks aside, what do you mean 'world of infinity'? That makes no sense.

Last edited by Kargoneth (2007-12-11 17:15:42)

JohnnyReinB · 2007-12-11 22:30:53

My point exactly...

LuisRodg · 2008-01-11 10:12:26

Could anyone explain the trick of "making the hole larger than the paper"? I do not understand it.

JohnnyReinB · 2008-01-11 17:53:28

you cut it zigzaggedly

i---------------i
l ________ l
l_______ l l
l ________l l
l l _______l
l l_______ l
'---------------'

If, you cut it like that, it would open the paper up, and now you created a hole with a larger size than the original paper.

John E. Franklin · 2008-01-12 02:25:39

Wow! I gotta try that! Be back...

John E. Franklin · 2008-01-12 02:32:58

That's totally awesome!!! The first one I made I missed the two side cuts in and I ripped it apart.
Then I looked at yours again and made a new one with the zigzag and two cuts off the sides.
It really does get like twice the size of a piece of paper!!! Hexagonal and twisty frame.

JohnnyReinB · 2008-01-12 18:12:22

In fact, the more times you cut it (by making the spaces between zigs and zags narrower) you can make it larger and larger, which returns us to the main topic of this, uhm, topic.

simron · 2008-03-15 14:54:08

Shall we assume that this is not 3-dimentional matter, but a 2-dimentional plane segment? Therefore, the limit of the percentage of the paper consumed by the hole (no zigzaggedy lines) is 100%. But it cannot be 100%. Unless you warp the quantum mechanical stuffage. Any tips?

RickyIsAnIdiot · 2009-06-17 22:43:49

For simron
Quote:

But it cannot be 100%. Unless you warp the quantum mechanical stuffage. Any tips?

RickyIsAnIdiot

The Hole! cannot be 100% So 99.999...% has to be the Size of the Hole! There must always be some Paper around the Outside!.

soroban · 2009-06-19 05:41:27

Hello, everyone!

Someone already explained the "trick" to this classic puzzle.

It is usually given in a variety of forms:
. . "Cut a hole in a standard playing card so that it will fit over your head."
. . "Cut a hole in a sheet of typing paper so that you can walk through it."

Fold the paper lengthwise.

      *-----------------------*
      |                       |
      |                       |
      |                       |
      |                       |
      |                       |
      * - - - - - - - - - - - * ← fold

.

Along the fold, make several cuts across the paper
. . but not all the way across.

      *-----------------------*
      |                       |
      |   |       |       |   |
      |   |       |       |   |
      |   |       |       |   |
      |   |       |       |   |
      * - * - - - * - - - * - *

.

Along the outside edge make several cuts between the first cuts,
. . again, not all the way across.

      *-------*-------*-------*
      |       |       |       |
      |   |   |   |   |   |   |
      |   |   |   |   |   |   |
      |   |   |   |   |   |   |
      |   |       |       |   |
      * - * - - - * - - - * - *

.

Cut along the fold between the first and last cuts.

      *-------*-------*-------*
      |       |       |       |
      |   |   |   |   |   |   |
      |   |   |   |   |   |   |
      |   |   |   |   |   |   |
      |   |       |       |   |
      * - *-------*-------* - *
          : ← ←  cut  → → :

.

Unfold the paper carefully.
You will have a "ring" of paper.

The ring can be made larger by applying more cuts.
.

RickyIsAnIdiot · 2009-06-20 00:55:36

For soroban above!...

Can you tell me what Percent the Hole is! and what Percent the Paper is!

RickyIsAnIdiot

Math Is Fun Forum

#26 2007-11-07 04:09:55

Re: Cut A Hole! In A Sheet Of Paper Problem!...

#27 2007-11-07 21:36:08

Re: Cut A Hole! In A Sheet Of Paper Problem!...

#28 2007-12-11 17:11:46

Re: Cut A Hole! In A Sheet Of Paper Problem!...

#29 2007-12-11 22:30:53

Re: Cut A Hole! In A Sheet Of Paper Problem!...

#30 2008-01-11 10:12:26

Re: Cut A Hole! In A Sheet Of Paper Problem!...

#31 2008-01-11 17:53:28

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#32 2008-01-12 02:25:39

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#33 2008-01-12 02:32:58

Re: Cut A Hole! In A Sheet Of Paper Problem!...

#34 2008-01-12 18:12:22

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#35 2008-03-15 14:54:08

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#36 2009-06-17 22:43:49

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#37 2009-06-19 05:41:27

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#38 2009-06-20 00:55:36

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