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The work surveys regional, national, and international contests, summarizes the reasoning behind each problem, and includes solutions that range from basic shape feasibility to more mathematically engaging constraints. 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It intentionally excludes many problems about nets of polyhedra and developments of cones or cylinders.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are the problems collected and organized?",{"text":118,"@type":114},"Problems are gathered by checking regional, national, and international competitions worldwide, then presented arranged by sub-topic. 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Ittherefore seems like an obvious step forme to have a closer look at problems in this area thathave been used as competition questions in various countries. Although the result can hardlybe called complete, what follows seems to be a fair representation of the kinds of ideas thathave been used in this way.  \nIn collecting these problems, I checked through various regional, national and internationalcompetitions from all around the world. I have intentionally focused on flat-folding problems,and have therefore not collected the many problems concerning nets of polyhedra ordevelopments of cones or cylinders that can be found in the literature. Such a collection couldmake for an interesting paper at some future point in time.  \nMost of the problems in this collection were set at lower levels, but many arenot at all as easyto solve as one might expect, and some are even mathematically quite interesting. Theproblems arepresented arranged by sub-topic, and together with the overview of problemsalready tobe found in the literature, for some of the sub-topics I also present some ideas forrelated potential competition problems.  \n## 1. SHAPES\n\nThe most elementary paperfolding problems are those concerning the shape (i.e. the outline)of a flat folded piece of paper. The medium tobe folded is almost always a rectangle or asquare, as these are the standard shapes in which paper is generally available. Inmost suchproblems, the paper is folded over only once. Sometimes the problems are embedded in quiteinnovative settings.  \n1) Three shapes X, Y and Z are shown below. A sheet ofA4 paper (297 mm by 210 mm)is folded once and placed flat on a table. Which of these shapes could be made?  \nX Y Z  \nA) Y and Z only B) Z and X only C) X and Y only D) none of them E) all of themsource: UK Junior Mathematical Challenge 1999, Nr. 19.  \nSolution: As we see in the figure, all are possible. The correct answer is therefore E.  \n2) Carla is looking out her window. She sees a rectangular flag movingin the wind. She copies the outline of the flag as it appears to her fivetimes. Which of the following pictures cannot be correct if we knowthat the flag was not torn?  \nA)  \nB)  \nC)  \nD)  \nE)  \nsource: Kangaroo 2020, Écolier  \nSolution: Since the flag isnot torn, the side connecting it to the flagpole can never appearshorter than it actually is. The correct answer is therefore B.  \n3) A sheet ofA4 size paper (297 mmx 210 mm) is folded once and then laid flat on thetable. Which of these shapes could not be made?  \nA)  \nB)  \nC)  \nD)  \nE)  \nsource: UK Intermediate Mathematical Challenge 1999, Nr. 2.  \nSolution: As wecan seein the figure, folding the rectangle along thediagonal does not yield the triangle shown in B. All other shapes arepossible, andthe correct answer is therefore B.  \n4) A piece of paper in the shape of a polygon is folded in half along aline of symmetry.The resulting shape is also folded in half, again along aline of symmetry. The finalshape is a triangle. How many possibilities are therefor the number of sides of theoriginal polygon?  \nA) 3 B) 4 C) 5 D) 6 E) 7source: UK Junior Mathematical Challenge 2007, Nr. 25.  \nSolution: If we consider unfolding the final triangle, we see that the edge of this triangle thatwas the crease isin the interior of the previous polygon. The other two edges ofthe finaltriangle have mirror images in the previous polygon, which can be extensions ofthe sides ofthe triangle. The intermediate polygon can therefore have at most 4 sides. Similarly, theoriginal polygon can have at most 6 sides. (In general, unfolding ann-gon yields an m-gonwith m 􀁤 2(n-1).) The possibilities forthe number of sides of the original polygon aretherefore 3, 4, 5 and 6. As we see in the following","cbCaic0YqYD6tHU1","https://ap.wps.com/l/cbCaic0YqYD6tHU1","pdf",559710,18,"English","# INTRODUCTION\n# 1. SHAPES\n## Problems about Paper","[{\"question\":\"What types of paperfolding problems does the paper focus on?\",\"answer\":\"It focuses on flat-folding problems using standard starting shapes such as rectangles or squares. It intentionally excludes many problems about nets of polyhedra and developments of cones or cylinders.\"},{\"question\":\"How are the problems collected and organized?\",\"answer\":\"Problems are gathered by checking regional, national, and international competitions worldwide, then presented arranged by sub-topic. For some sub-topics, the author also suggests ideas for related potential competition problems.\"},{\"question\":\"What kinds of reasoning appear in the solutions?\",\"answer\":\"Solutions use geometric constraints from the folding process, such as which outlines are feasible after folding once and how creases act as edges or mirror-image boundaries in unfolded or intermediate configurations.\"}]","Folding Questions - A Paper about - Problems about Paper | PDF",1790501323,45]