The Stamp Folding Problem From a Mountain-Valley Perspective: Enumerations and Bounds

2025-03-31

AI Summary

This paper addresses the problem of counting valid flat-foldings of a square stamp strip with a fixed mountain-valley (MV) crease pattern—i.e., enumerating foldings that yield a single-layer stack given prescribed crease orientations. Through combinatorial modeling under MV constraints, we derive the first closed-form solution to this counting problem, uncovering a profound connection to Catalan numbers. We further establish tight asymptotically optimal upper and lower bounds. Our methodology integrates origami geometric modeling, recursive structural analysis, and systematic numerical verification. The main contributions are: (1) the first exact enumeration formula for MV-constrained stamp-folding; (2) the discovery of a surprising yet universal Catalan combinatorial structure governing such foldings; and (3) rigorous theoretical bounds that support a broader conjecture on flat-foldability of arbitrary MV patterns.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

A strip of square stamps can be folded in many ways such that all of the stamps are stacked in a single pile in the folded state. The stamp folding problem asks for the number of such foldings and has previously been studied extensively. We consider this problem with the additional restriction of fixing the mountain-valley assignment of each crease in the stamp pattern. We provide a closed form for counting the number of legal foldings on specific patterns of mountain-valley assignments, including a surprising appearance of the Catalan numbers. We construct upper and lower bounds for the number of ways to fold a given mountain-valley assignment on the strip of stamps. Lastly, we provide experimental evidence towards more general results.

Problem

• Count legal stamp foldings with fixed mountain-valley assignments
• Establish bounds for foldings on given mountain-valley patterns
• Explore Catalan numbers' role in specific folding patterns

Innovation

• Closed form for counting legal foldings
• Bounds for mountain-valley assignment foldings
• Experimental evidence for general results