# No. Of ways of folding a paper strip

You have given a strip which is divided into n+1 identical parts by n folds on the strip. Now find the no. Of ways in which you can fold the whole strip into a single identical part..!

• This was a problem in Martin Gardner's column in Scientific America decades ago. – John Wayland Bales Mar 31 at 18:56
• Here is a discussion of folding a strip of stamps. – John Wayland Bales Mar 31 at 18:59
• But he doesn't provide answer to my question about – user647127 Mar 31 at 19:20
• See OEIS A000136 – Ross Millikan Mar 31 at 21:04
• But he does make the claim "There seems not to be a closed-form formula to compute the counts of labeled stamp foldings, though many have tried." – John Wayland Bales Mar 31 at 23:23

With the restriction imposed by $$n$$ folds producing $$n+1$$ identical parts (folding the strip in half and in half again results in $$4$$ identical parts but there are actually $$3$$ folds in the strip's single thickness), then the number of ways to fold the strip into identical parts is $$2^n$$. Conceptually, each fold can go in one of two directions so the $$2$$ possibilities of the first fold is followed by $$2$$ possibilities for the next fold etc. Each resulting configuration of folds has a similar configuration with surface A flipped with surface B. There are practical limitations of actually doing this for some folding configurations as n gets large.
• But for $n=3$ you can have mountain folds between $1$ and $2$ and between $2$ and $3$, then a valley fold between $3$ and $4$. The squares can then be stacked $2134$ or $2341$ I believe these should be counted as distinct. – Ross Millikan Mar 31 at 21:01