6.1.1993

Folding stamps

 

I found the idea in

 

Michael Holt, "Neue mathematische Aufgaben für Denker und Tüftler", Studio Dumont 1988, Seite 27.

 

In how many ways n connected stamps (in a line) can be folded?

 

The answer for 2, 3 and 4 is 1, 2 and 5 resp. See the illustration.

 


 


5 stamps can be folded in 14 ways:

 


 

 


6 stamps can be folded in 38 ways:

 


 

 


Finding the number of foldings for bigger n’s must be done by a programm which can have the following steps

 

1)      Scan the list of all n! permutations of  1 2 3 .. n

2)      For each horizontally written permutation link 1 with 2 on top. This separates de figures in different groups.

3)      Then link 2 with 3 on bottom

4)      Now check if 3 and 4 are in the same group (above). If not the permutation has failed and we try the next permutation. Otherwise link 3 with 4 and update the grouping.

5)      Continue beneath with figures 4 and 5.

6)      and so on

7)      until n is linked to  n-1. If yout get here you have a solution.

 

Let’s name the number of collected solutions in this way 'brutto'.

 

Now we have to eliminate multiple solutions. We consider that solution B which you get from solution A by flipping top/down is not different.

 

We also consider that solution C which you get from solution A by turning the strip of stamps by 180° first and applying the same folding sa for solution A afterward is not different.

 

Also a combination of B and C is not different.

 

Let’s name this reduced folding number 'netto'.

 

Here is a table with the numbers for one to 10 stamps

 

n

n!

brutto

netto

2

2

2

1

3

6

6

2

4

24

16

5

5

120

50

14

6

720

144

38

7

5040

462

120

8

40320

1392

353

9

362880

4536

1149

10

3628800

14060

3527

 

 

The ratio brutto/netto which measures the multiplicity of solutions approaches 4 for big n’s.

 

Brutto andd netto are growing by a factor of three for big n’s.

 

The function brutto(n)/brutto(n-1) shows an alternating pattern. It is clear that the possible symmetries are different if n is even or odd. If we separate the even and odd points to form two different curves, we can nicely extrapolate for the 11 and 12 stamps and get approxiamtly 11’500 and 36’400 netto foldings resp.