台本（折り紙）, English version 2

A numbered strip

Please take your scissor, and cut out a strip of paper
numbered 1,2,3, and 4.

OK let's review a bit about high school mathematics.
Do you know how many ways of arranging four numbers 1,2,3,4
in line.

Yes it is the numbers of permutations of the numbers, and
it is the fractorial of 4, denoted by 4 !(階乗),
which equals to 24. This is the numbers of permutations
（順列）of 1,2,3,4.

(Write down the 24 patterns on the black board.
An important idea here is exhaustion. )

By folding the strip you can pile up the numbers 1 to 4.
For example, by folding in this manner you can obtain,
the piling pattern 1,2,3,4 form top to bottom.

Then let me ask you a question.
The question is: among the 24 patterns of the permutations,
which one can be realized by folding the strip ?

Please try, and tell me if you can realize it.
I will check the pattern on the board.

For example, if you consider the pattern 1423.
This pattern seems not to be able to realize.
Is there anyone who can make an explanation of the fact ?
(A key is to consider the cross section.)


You first place the numbered segments in that order
from top to bottom.

−−−−−１−−−−−−−
−−−−−４−−−−−−−
−−−−−２−−−−−−−
−−−−−３−−−−−−−


Then try to join  endpoints of the segments with
short vertical arcs according to the order given by
the numbers.

OK, then here is a 2-dimensional version of the question.

Please take a look at the pattern.
Can you fold the pattern with following the pattern,
of course not tiering the paper ?

Actually this is related to the problem:
Find an efficient way of deciding
whether it can be folded or not.
This problem is not solved yet, which means that
it is a frontier of Mathematics.