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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..!

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  • $\begingroup$ This was a problem in Martin Gardner's column in Scientific America decades ago. $\endgroup$ – John Wayland Bales Mar 31 at 18:56
  • $\begingroup$ Here is a discussion of folding a strip of stamps. $\endgroup$ – John Wayland Bales Mar 31 at 18:59
  • $\begingroup$ But he doesn't provide answer to my question about $\endgroup$ – user647127 Mar 31 at 19:20
  • $\begingroup$ See OEIS A000136 $\endgroup$ – Ross Millikan Mar 31 at 21:04
  • $\begingroup$ 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." $\endgroup$ – John Wayland Bales Mar 31 at 23:23
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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.

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    $\begingroup$ 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. $\endgroup$ – Ross Millikan Mar 31 at 21:01
  • $\begingroup$ @Ross Millikan I see what you are saying, for identical folds the strip can be stacked in different ways. The way I interpreted the question was simply a fold configuration. For n=3, this would result in 12 configurations. I'll look to see if there is a pattern for increasing n. Thanks $\endgroup$ – Phil H Apr 1 at 14:44

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