On stretch minimization problem on unit strip paper

For a given mountain-valley pattern of equidistant creases on a long strip paper, there are many folded states consistent with the pattern. Among these folded states, we like to fold a paper so that the number of the paper layers between each pair of hinged paper segments is minimized. We first formalize this problem as optimization problem. The complexity of the problem is not known. In this paper, we give partial results related to the problem. First, we show that the problem is well-defined even in a simple folding model. The simple folding model is the most primitive model of basic origami models, and hence the folding availability is very restricted. We show a universality theorem of the simple folding model for this problem. That is, every flat folded state consistent with any given pattern can be folded by a sequence of simple foldings. Next, we investigate the number of folded states consistent with a given pattern. For a given random mountain-valley pattern, the expected number of folded states consistent with the pattern is exponential.