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Cutting Slits in a Piece of Paper


Date: 02/01/2002 at 10:01:23
From: Scottie Plyler
Subject: Hole in paper

How can you cut a hole in a piece of paper so a person can go through 
it?


Date: 02/01/2002 at 12:09:37
From: Doctor Ian
Subject: Re: Hole in paper

Hi Scottie,

Take a piece of paper.  Fold it in half:

   +---------------------+  <--- edge of fold
   |                     |
   |                     |
   |                     |
   |                     |
   +---------------------+



Near the left and right sides, and in the middle, cut a slit from the 
folded edge almost, but not quite, to the other edge:

   +--.-------.-------.--+  <--- edge of fold
   |  .       .       .  |
   |  .       .       .  |
   |  .       .       .  |
   |  .       .       .  |
   |                     |
   +---------------------+


Now cut two slits in between these slits, coming from the other 
direction. Again, don't go all the way to the fold:

   +--.-------.-------.--+  <--- edge of fold
   |  .       .       .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |      .        .     |
   +------.--------.-----+


Now cut across the fold, except at the very ends:

   +--.................--+  <--- edge of fold
   |  .       .       .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |      .        .     |
   +------.--------.-----+



Before making this last cut, if you unfold the paper, it should look 
like this:

   +------.--------.-----+
   |      .        .     |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .       .       .  |
   |  .       .       .  | <------ edge of fold
   |  .       .       .  |  
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |      .        .     |
   |      .        .     |
   +------.--------.-----+


So the last cut goes here:

   +------.--------.-----+
   |      .        .     |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .       .       .  |
   |  .................  | <------ edge of fold
   |  .       .       .  |  
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |      .        .     |
   |      .        .     |
   +------.--------.-----+


(It's just easier to make the cut while the paper is still folded.)  

Now, if you grab the two points labeled A and B, 

   +------.--------.-----+
   |      .        .     |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |  .       .       .  |
 A |  .................  | B 
   |  .       .       .  |  
   |  .   .   .    .  .  |
   |  .   .   .    .  .  |
   |      .        .     |
   |      .        .     |
   +------.--------.-----+

and gently pull them apart, you should find that you've made a 
surprisingly large hole in the paper! By making more slits, you can 
make the hole even bigger:


   +-.---.---.---.---.-+ <--- edge of fold
   | .   .   .   .   . |
   | . . . . . . . . . |
   | . . . . . . . . . |
   | . . . . . . . . . |
   | . . . . . . . . . |
   |   .   .   .   .   |
   +---.---.---.---.---+


Cut enough slits, and you could drive a _car_ through the hole. 

I hope this helps.  Write back if you'd like to talk more about this, 
or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Geometry
Elementary Puzzles
Elementary Two-Dimensional Geometry
Middle School Geometry
Middle School Puzzles
Middle School Two-Dimensional Geometry

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