Mathematical Stories I – Graphs, Games and Proofs: For Gifted Students in Primary School (essentials) [1st ed. 2021] 3658327324, 9783658327323

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Mathematical Stories I – Graphs, Games and Proofs: For Gifted Students in Primary School (essentials) [1st ed. 2021]
 3658327324, 9783658327323

Table of contents :
What You Can Find in This essential
Preface
Contents
1 Introduction
1.1 Mathematical Goals
1.2 Didactic Notes
1.3 The Narrative Framework
Part I Tasks
2 Colorful Mathematics
3 Yet Another Task Without a Solution
4 Dangerous Game Against a Dragon
5 Revenge: A New Game Against the Dragon
6 Word Puzzles and Graphs
7 Even More Word Puzzles and Graphs
Part II Sample Solutions
8 Sample Solution to Chapter 2
9 Sample Solution to Chapter 3
10 Sample Solution to Chapter 4
11 Sample Solution to Chapter 5
12 Sample Solution to Chapter 6
13 Sample Solution to Chapter 7
What You Learned From This essential
References

Citation preview

Susanne Schindler-Tschirner Werner Schindler

Mathematical Stories I – Graphs, Games and Proofs For Gifted Students in Primary School

essentials

Springer essentials

Springer essentials provide up-to-date knowledge in a concentrated form. They aim to deliver the essence of what counts as "state-of-the-art" in the current academic discussion or in practice. With their quick, uncomplicated and comprehensible information, essentials provide: • an introduction to a current issue within your field of expertis • an introduction to a new topic of interest • an insight, in order to be able to join in the discussion on a particular topic Available in electronic and printed format, the books present expert knowledge from Springer specialist authors in a compact form. They are particularly suitable for use as eBooks on tablet PCs, eBook readers and smartphones. Springer essentials form modules of knowledge from the areas economics, social sciences and humanities, technology and natural sciences, as well as from medicine, psychology and health professions, written by renowned Springer-authors across many disciplines.

More information about this subseries at http://www.springer.com/series/16761

Susanne Schindler-Tschirner · Werner Schindler

Mathematical Stories I – Graphs, Games and Proofs For Gifted Students in Primary School

Susanne Schindler-Tschirner Sinzig, Germany

Werner Schindler Sinzig, Germany

ISSN 2197-6708 ISSN 2197-6716 (electronic) essentials ISSN 2731-3107 ISSN 2731-3115 (electronic) Springer essentials ISBN 978-3-658-32732-3 ISBN 978-3-658-32733-0 (eBook) https://doi.org/10.1007/978-3-658-32733-0 © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Iris Ruhmann This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

What You Can Find in This essential

• • • • •

Learning units in stories Graphs Mathematical games Proofs Sample solutions

v

Preface

The conception and design of this essential and of Volume II of the “Mathematical Stories” (Schindler-Tschirner and Schindler 2019) are the result of the experience gained from a mathematics study group for gifted students, which the second author led at the primary school in Oberwinter (Rhineland-Palatinate). Twelve students from grades 3 and 4 took part. This was 10% of all students in these two grades. Of these, at least1 three students were later able to win prizes in national mathematics competitions. Of course the authors do not assume that these successes were only made possible by participation in this mathematics study group. Rather, with the two essentials, we would like to make a contribution to awakening interest in and enjoyment of mathematics and promoting mathematical talent. Sinzig January 2019

1 The

Susanne Schindler-Tschirner Werner Schindler

authors are no longer in contact with all participants of the mathematics study group.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Didactic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Narrative Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1 1 4 5

Tasks

2

Colorful Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

3

Yet Another Task Without a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4

Dangerous Game Against a Dragon . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

5

Revenge: A New Game Against the Dragon . . . . . . . . . . . . . . . . . . . . . .

19

6

Word Puzzles and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

7

Even More Word Puzzles and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Part II

Sample Solutions

8

Sample Solution to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

9

Sample Solution to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

10 Sample Solution to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

11 Sample Solution to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

12 Sample Solution to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

13 Sample Solution to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 ix

1

Introduction

Apart from the introduction, this essential consists of two parts, each with six chapters. Part I contains tasks and Part II the sample solutions discussed in detail with didactic suggestions, mathematical goals and outlooks. This book and the follow-up volume (Schindler-Tschirner and Schindler 2019) are aimed at leaders of study groups (SGs) and support courses for mathematically gifted students in grades 3 and 4, at teachers who practice differentiated mathematics lessons, and also at committed parents for extracurricular support. The sample solutions are tailored to the leadership of SGs; modified accordingly, they can also serve as a guide for parents who work through this book together with their children.

1.1

Mathematical Goals

This essential is intended to convey the joy of mathematics as well as the insight that mathematics does not only consist of learning more or less complicated “formulas.” It differs fundamentally from some pure task collections, which contain interesting and by no means trivial mathematical tasks “to puzzle with,” but in which, in our view, the targeted learning and application of mathematical techniques is not considered enough. The mathematical talent of primary school children and its promotion has played an important role in primary school pedagogy for several decades. This book does not go into general didactic considerations and theories on the promotion of gifted children in detail, although the bibliography contains a selection of relevant publications for the interested reader. This essential focuses on the tasks, the applied mathematical methods and techniques and on concrete didactic suggestions for implementation in a gifted SG. This essential does not require a special mathematics textbook in primary school. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_1

1

2

1

Introduction

In school lessons, high-performing students usually need little perseverance to solve the mathematical problems assigned. This is usually quite “straight forward,” and often the children get bored after a short time. It did not seem to make much sense to the authors to discuss only slightly more complicated problems than in school lessons. Instead, this book contains tasks that have hardly any comparable examples in normal school lessons and which encourage the mathematical thinking of the children. Even talented students will hardly find tasks that they can solve “just like that.” In this respect, the tasks also present a new challenge for them. The students are guided in the task chapters to work out the solutions as independently as possible (but with the help of the instructor!). The solution of the tasks requires a high degree of mathematical imagination and creativity, which is encouraged by dealing with mathematical problems. Unlike in the following volume (Schindler-Tschirner and Schindler 2019), this volume does not “calculate,” which is the first surprise for the children. Thus, the students very quickly make the experience that mathematics is more than just arithmetic. Chapter 2 deals with path problems, some of which have no solutions. To prove this, first the relevant information has to be extracted from a city map and modeled by a graph. Finally, a coloring proof is used to show that in fact no solutions can exist. Graphs and, of course, the use of a strictly logical mathematical proof is new territory for children. This is first of all “heavy fare,” but it also gives first insights into what is important in mathematics. In Chap. 3, the path problems from Chap. 2 are taken up first, but with a slightly modified city map. Hence not only the proof from Chap. 2 collapses, but it also makes the statement wrong. The students learn that even small changes in the preconditions can have considerable effects. The rest of Chap. 3 deals with coverage problems. With a further coloring proof, it is shown that a given task cannot have a solution. In Chap. 4, a mathematical game is systematically analyzed and the optimal game strategy is determined. The students learn how to trace a game back to simpler variants of the same game and thus determine the winning strategy. In Chap. 5, a similar game is analyzed. By adapting the strategy to another game, the procedure from Chap. 4 is practiced again and the understanding is deepened. Chap. 6 also deals with a real world problem, namely word puzzles. The question arises how often a certain word occurs in a honeycomb structure. First, a modeling by a directed graph is carried out, and then the task is simplified step by step until the solution is achieved. The procedure is methodically not so easy, and therefore Chap. 7 ties in directly with Chap. 6. This gives the children the opportunity to rehearse and deepen what they have learned. The result is concrete numbers, which the children are naturally familiar with. This will also give the children an additional sense of achievement.

1.1 Mathematical Goals

3

Table II.1 shows the mathematical techniques that are learned in the individual chapters. For further success in mathematics, it is essential to develop, try out and modify your own ideas. Of considerable importance is the ability to recognize what has already been learned in a modified form (“Where have you seen this before?”). A certain tolerance of frustration is also indispensable, i.e., to “cope” with unsuccessful approaches to solutions and to keep developing and pursuing new ideas. Added to this are “soft skills” such as patience, perseverance and tenacity. These qualities are also trained by working with the tasks of this essential; see also Sects. 13.3 and 13.6 in Käpnick (2014). In this respect, the tasks of this book even provide first experiences that, if one looks very far into the future, are also helpful for a possible later study of mathematics, computer science or natural and engineering sciences. Besides the solution of the tasks themselves, the focus is also on the mathematical methods used. The mathematical methods and techniques learned are also used extensively in mathematics competitions in the primary and middle school levels (and occasionally even in the upper school level), such as the annual Mathematical Olympiad with class-specific tasks from grade 3 (Mathematical Olympiads e. V. 1996–2016, 2013b, 2017–2018), the Federal Mathematics Competition (Langmann et al. 2016) and the Fürth Mathematics Olympiads (Verein Fürther Mathematik-Olympiade e. V. 2013; Jainta et al. 2018), to name but a few. For the interested reader, the bibliography contains a number of other books with tasks and solutions from national and international mathematics competitions as well as collections of tasks, but these are mostly aimed at older students. The kangaroo competition (Noack et al. 2014) plays a special role here due to its task structure (multiple choice). The task collections Schiemann and Wöstenfeld (2017) and (2018) contain a selection of the most interesting tasks from the annual school competition “Maths in Advent,” which takes place in December each year and was initiated by the German Mathematical Society (Deutsche MathematikerVereinigung, DMV) in 2008. The tasks are of varying degrees of difficulty and are intended to appeal to a wide range of interested students. Monoid, a mathematics journal for students at school, published by the University of Mainz (Institute of Mathematics at Johannes-Gutenberg University of Mainz 1981–2019) should also be highlighted. We should also refer to Beutelspacher (2005), Enzensberger (2018) and Beutelspacher und Wagner (2010), which combine mathematics with fiction in an entertaining way and invite readers to browse and experiment with mathematics. It corresponds to the experience of the two authors that in supra-regional mathematics competitions from the middle school upward there is usually an

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1

Introduction

accumulation of participants from a few schools. Often, interested students are offered targeted support through mathematics SGs or other measures. As former scholarship holders of the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes), both authors are particularly interested in promoting gifted students. Both essentials contain carefully prepared learning units with detailed sample solutions for mathematics SGs for gifted students in primary schools. In this way, we would like to make a contribution to the promotion of gifted children in primary school. In addition to the mathematical content, we want to awaken the students’ enjoyment of mathematics and encourage them to make mathematical discoveries.

1.2

Didactic Notes

Part II contains detailed sample solutions for the tasks from Part I with didactic hints and assistance for the implementation in a SG. The solutions shown are designed in such a way that they are understandable and comprehensible even for non-mathematicians. The sample solutions are not directly intended for children. In addition, the mathematical goals of the respective chapters are explained and outlooks are given where the mathematical techniques learned are applied in mathematics and computer science. The abilities of the participating students should not be underestimated, and also not overestimated. It is important to explain (repeatedly) to them from the beginning that even very good students are not expected to be able to solve all tasks independently. This is very important, because a permanent overtaxing and/or (perceived) failure can lead to lasting frustrations, which are certainly not conducive to the attitude toward mathematics. This would be the opposite of what this essential wants to achieve. Therefore, participants should be carefully selected. In the mathematics SG mentioned above, the participants were suggested by the class teachers of grades 3 and 4. Chapters 2 to 7 consist of many subtasks, the level of difficulty of which usually increases. Weaker students should preferably work on the easier subtasks. Some of the subtasks are very well suited for working in small groups of 2 to 3 students. This is sometimes pointed out in the sample solutions. The teacher should give the students enough time to discover their own solutions and to pursue approaches that do not correspond to the sample solutions. It is not easy, if not impossible, to develop tasks that are optimally tailored to the needs of any mathematics SG or support course. It is at the discretion of the course instructor to leave out subtasks or add his/her own subtasks. In this way,

1.3 The Narrative Framework

5

he or she can influence the level of difficulty to a certain extent and adapt it to the abilities of the course participants. The students’ grasp and understanding of the solution strategies should in any case be given priority over the goal of “completing” as many subtasks as possible. The individual chapters should normally require more than one course meeting. If the teacher works with task sheets, they should be read together, but only those subtasks that are next to be worked on. Presenting all the subtasks at once could lead to rapid discouragement and resignation among the participants right from the start. A proven procedure is to have a high-performing student read the text out loud and, if necessary, clarify the task. Since younger students participate in the SG, this step is very important. Problems of understanding the tasks should not be underestimated. Normally you should start with task part (a). Each student should have a reasonable amount of time (depending on the level of the learning group) to think about the task alone (with help if necessary). Afterwards the different ideas, approaches or perhaps even finished solutions are collected. Each student should regularly be given the opportunity to present his or her approach or solution to the others. This not only allows the students to reflect on their own approach again, but also to practice such important skills as a clear presentation of their own considerations and mathematical argumentation; see also Nolte (2006, p. 94). In the mathematics SG mentioned above, fourth-graders were on average significantly better performers than third-graders. This was not because the third graders lacked the necessary previous mathematical knowledge. Rather, this was probably the result of greater intellectual maturity among the older students. This may not be surprising for experienced teachers. In any case, the instructor should keep an eye on this effect. The embedding of the tasks in a large, continuous adventure story not only forms the narrative framework, but also gives the children a feeling of security. At the beginning of each new lesson, the instructor brings the children back into the fairy-tale, enchanted world of Clemens, so that fear of contact with the tasks does not even arise.

1.3

The Narrative Framework

Anna and Bernd are in the third grade. Their favorite subject is mathematics, and they are pretty good at it. They are eager to join the club of enthusiastic young mathematicians, or, to put it more briefly, the CoEYM (Fig. 1.1). Unfortunately,

6

1

Introduction

Fig. 1.1 Emblem of the CoEYM

n · (n+1) : 2

according to the club’s statutes, one may only join the CoEYM if one attends at least the fifth grade. So far there have been no exceptions. However, Anna and Bernd are very persistent, and finally the club chairman Carl Friedrich makes them an offer: “Well, I’ll give you a chance to join the CoEYM right now. But first you have to prove that you deserve this special treatment. To do so, you have to help the sorcerer’s apprentice Clemens to pass 12 mathematical adventures.1 To become a real wizard, Clemens must earn a number of magic items (e.g., a wand or a bit of dragon ointment) by solving difficult math problems. Just so you know: Clemens is the club mascot of the CoEYM.” And he adds: “Let me make one thing clear: You will either be admitted together or not at all. So you must learn to solve mathematical problems together.” Then Carl Friedrich grins a little, turns away and walks away. He absolutely cannot imagine that Anna and Bernd pass this entrance examination. Could he be wrong?

1 Six

adventures each in this essential (Chapters 2–7) and in Volume II.

Part I Tasks

There are six chapters with tasks. In the narrative context, these are the mathematical adventures of sorcerer’s apprentice Clemens. New mathematical terms and techniques are introduced. The story and the tasks (and of course the instructor!) lead the children to the right way of solving the problems. Each chapter ends with a section that describes the current situation from the perspective of Anna, Bernd and Clemens. With a short summary of what the students have learned in this chapter, this section emerges from the narrative framework at the end. This description is not given in technical terms as in Table II.1, but in language suitable for the students.

2

Colorful Mathematics

Sorcerer’s apprentice Clemens lives in Right Angleton. Clemens wants to buy a magic wand in the magic shop there. “It’s not that easy,” hums Mercator Magicus, the owner of the magic shop. “A magic wand is only given to someone who has solved tricky math problems before. Where do you live, Clemens?” “At 13 Angle Lane,” answers Clemens. “Well, I can think of some very interesting mathematical problems right away,” replies Mercator Magicus, digging a map of Right Angleton out of a drawer and drawing in Clemens’ house and the magic shop. You have to know that the streets in Right Angleton only run in an east–west or in a north–south direction and intersect at right angles. The connection between two intersections (intersections of streets) is called a street section. From each intersection, Clemens can go to any neighboring intersection. Figure 2.1 shows a small section of Right Angleton. Because it goes on in all directions. “I’d like you to tell me how you can get from your home to my magic shop.” “But that’s easy,” Clemens triumphs. “After all, I did find my way here.” “But it’s not that easy to get a magic wand,” hums Mercator Magicus. “You are supposed to find paths of given length.” Clemens drives with his right index finger along the city map and quickly finds some paths. But this is not enough for Mercator Magicus. Clemens should write down the paths. a) Think about how you can write down paths. After Clemens has overcome the first hurdle, Mercator Magicus gives him four tasks. b) Find paths from your (Clemens) house to the magic shop, which are 5 street sections long. Write down these paths. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_2

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Colorful Mathematics

Clemens

magic shop Fig. 2.1 City map of Right Angleton (small section)

c) Find paths from your (Clemens) house to the magic shop, which are 6 street sections long. Write down these paths. d) Find paths from your (Clemens) house to the magic shop, which are 7 street sections long. Write down these paths. e) Find path from your (Clemens) house to the magic shop, which are 8 street sections long. Write down these paths. Clemens quickly finds some paths of length 5 and 7, but he does not make progress with tasks (c) and (e). He simply does not find suitable paths. For a short moment, he thought he had found a path of eight street sections, but unfortunately he just miscounted. Clemens is totally disappointed. Does he have to go home without a magic wand? “That’s mean! There are certainly no solutions with 6 or 8 street sections,” he mumbles sullenly and wants to leave the magic shop already. “Stop,” cries Mercator Magicus. “In mathematics, one should not give up so easily. These two tasks are not easy,” says Mercator Magicus. “So I’ll help you a

2

Colorful Mathematics

11

Fig. 2.2 Undirected graph (example)

little. Maybe there really aren’t any paths of 6 or 8 street sections. If that really is the case, you have to prove it. Then you would have solved the two tasks.” But Clemens doesn’t know what proof is. “You have to show that there can be no solution. You haven’t done that yet. You just haven’t found a path.” “First simplify the city map and remove the unimportant so that you can see the essential,” says Mercator Magicus. f) Can you help Clemens with that? What can be left out of the city map (Fig. 2.1) and what is really important for tasks (c) and (e)? “The little houses on the city map are certainly not important for the solution,” says Clemens. “I’ll definitely leave them out.” After a while he shows the simplified city map to Mercator Magicus. “You are making progress,” Mercator Magicus praises. “Do you know what a graph is?” “No, we didn’t learn that in school. I’m sure I always paid attention!” Clemens adds hastily. “I believe you,” laughs Mercator Magicus. Mercator Magicus explains: “An undirected graph consists of vertices and edges that connect vertices. Figure 2.2 shows an example. There you see small circles connected by edges. The small circles are called vertices, by the way. You can think of the edges as streets that you can drive on in both directions to get to the adjacent vertices.”

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2

Colorful Mathematics

g) Simplify the city map again and display it as an undirected graph. Here, the street centers (of the intersections) are the vertices and the streets are the edges. After Clemens has completed this task, Mercator Magicus gives one last tip: “If you paint the vertices in the graph appropriately, you can prove that there are indeed no paths of 6 or 8 street sections”. After some consideration Clemens asks: “How should I color the vertices? I have no idea!” “Alright,” smiles Magister Magicus: “Choose two colors and color the vertices like a chessboard. But this is my last tip.” h) Help Clemens to prove that there are no paths from his house to the magic shop of 6 or 8 street sections. Clemens has now found the solution. “Proving is a great thing! But now I’d like to have a magic wand.” “Not yet. First you have to solve two more problems. But if you really understood the solution, it won’t be difficult anymore.” i) Is there a path that is 99 street sections long? j) Is there a path that is 2,020 street sections long? Anna, Bernd, Clemens and the Students Clemens is happy and proud that he has successfully completed his first mathematical adventure and now has his own magic wand. And how are Anna and Bernd? Anna says: “That was quite difficult! Fortunately, we were able to take advantage of the hints of the magic shop owner after some thinking and trying. Proving is a great thing, and I haven’t heard anything about graphs yet either. I’m already looking forward to the next adventure.” “Me too,” says Bernd. What I have Learned in this Chapter • • • • • •

Mathematics is not just calculating. I now know what an undirected graph is. Graphs can help us to solve practical tasks. Not every task has a solution. In mathematics, there are proofs. Skillful coloring can provide a proof.

3

Yet Another Task Without a Solution

Clemens passes the magic shop again and looks curiously into the shop window. There he sees a magic cloth with which you can make things invisible. “I absolutely must have that,” Clemens immediately thinks. “But for that I must surely first solve a mathematical problem. I can do it for sure.” Clemens pushes down the handle of the heavy glass door to the magic shop. Mercator Magicus comes out from behind a kind of counter and laughs: “Well, well, you want to have the magic cloth! A good choice. The magic cloth is in great demand. Let’s see what tasks it requires.” And after taking out some stacks of paper, he pulls out a sheet of paper and explains: “Imagine that an additional street is being built in Right Angleton, running diagonally. Then the place name Right Angleton would actually no longer be justified, but that doesn’t matter at the moment. Look at this city map (Fig. 3.1). There you can see what I mean. An additional street runs diagonally through a block of houses (fifth from the left, third from above).” a) Does the additional street change the solutions from the first mathematical adventure? So: Are there paths from Clemens’ house to the magic shop in 5, 6, 7, 8, 99 or 2,020 steps? Clemens soon finds the solution and proudly stretches out his hand toward the magic cloth. “Stop!” says Mercator Magicus. “That was just for warming up. A magic cloth is something very special. You have to solve two more tasks. The second is not so easy.” With a magic cloth, you can make exactly two adjacent squares (horizontal or vertical) invisible. Clemens should place 31 magic cloths on the chessboard (Fig. 3.2) in such a way that

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_3

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3 Yet Another Task Without a Solution

Clemens

magic shop Fig. 3.1 Slightly modified city map of Right Angleton (small section)

a b c d e

Fig. 3.2 Chessboard with inscription

f

g h

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1 a b c d e

f

g h

3 Yet Another Task Without a Solution

15

b) only the squares a1 and b3 remain visible c) only the squares a1 and h8 remain visible Clemens is supposed to give a solution or to show (prove) that there is no solution. Anna, Bernd, Clemens and the Students Clemens now owns a magic wand and a magic cloth. Bernd says to Anna: “Tasks that cannot be solved at all are something completely new. If you have to prove that there is no solution, it can be more difficult than giving a solution if the task is solvable. Proving is really cool! We haven’t learned that in class yet.” What I have Learned in this Chapter • A small change in the task can have a big impact on the solution. • I have again provided a proof that a task has no solution.

4

Dangerous Game Against a Dragon

In the library in Right Angleton, department “Mysteries,” Clemens discovers an old magic book. He pulls the book from the shelf, sits down at one of the large tables and begins to leaf through it. He learns about a magic ruby. But it is guarded by a fire-breathing dragon. The magic ruby is only given to the one who defeats the dragon in the dragon game. But whoever loses at the dragon game must serve the dragon for 99 years. Encouraged by the two successful adventures, Clemens sets off on his journey. Once he reaches the dragon, he learns the exact rules of the game. Dragon Game (Game Rules) On the table are 24 pieces of lava. The two players take alternately 1, 2 or 3 lava pieces away. Whoever succeeds in taking the last lava piece away wins the game. Clemens gets to decide who starts the game. What should he do? Is there a strategy he should follow? Unfortunately, Clemens has no idea how to proceed. “Time for my nap,” the dragon hums. “Afterward we’ll play, and from tomorrow on I have a new servant.” Clemens is quite anxious now. Can you help him? Fortunately, the dragon sleeps at least an hour. So we have enough time to approach the problem systematically. a) Play a few games of dragon game with your table neighbor to familiarize yourself with the dragon game. b) First, investigate simpler variants of the dragon game, where at the beginning there are only 1, 2, 3 or 4 pieces of lava on the table. Is it convenient to start the game? c) Investigate simpler variants of the dragon game, where at the beginning there are 5, 6, 7 or 8 pieces of lava on the table. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_4

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Dangerous Game Against a Dragon

We are lucky! The dragon is still sleeping, and now we can analyze another small game variant before we get to the real dragon game. d) Investigate the variant of the dragon game with 12 lava pieces. Take advantage of what you have learned in (b) and (c). After this preparatory work, the time has come to deal with the real dragon game. e) Now examine the real dragon game with 24 lava pieces. Clemens has understood through the parts (b)–(d) what is important in the dragon game. After the dragon has woken up, the dragon game can begin. “Start already, Clemens,” says the dragon cunningly. “No, I get to decide who starts. And that is you,” answers Clemens. Shortly afterward everything is over. Clemens has won the magic ruby. Anna, Bernd, Clemens and the Students Clemens now also owns a magic ruby, but that was pretty close. If the dragon had woken up just a few minutes earlier, Clemens would probably have become servant of the dragon for the next 99 years. Not a nice idea! Bernd says: “I didn’t know that playing games had anything to do with mathematics.” “Yes, but then playing is not only fun, but also exhausting,” Anna adds. What I have Learned in this Chapter • • • •

Mathematics also deals with games. A mathematical game has nothing to do with passing time. Instead, you look for the optimal game strategy. I have learned how to transform a difficult task step by step into simpler tasks until the solution is found.

5

Revenge: A New Game Against the Dragon

Clemens has successfully completed the last adventure, and the magic ruby now belongs to him. The dragon finds this very bad and offers Clemens a revenge. For this, the rules of the game are changed. Super Dragon game (Game Rules): There are 24 pieces of lava on the table. The two players alternately take away 1, 2, 3 or 4 lava pieces. Whoever takes away the last lava piece has lost the game. If Clemens loses, he does not have to serve the dragon for 99 years, but he has to give back the magic ruby. If Clemens wins, he will also receive a pinch of dragon ointment, which, as is well known, significantly enhances many spells. Clemens may again decide who starts the game. He ponders for a moment: “Should I accept the revenge and risk my ruby?” After a short hesitation, Clemens agrees to revenge. The game is played as soon as the dragon wakes up from its nap. Until then there is at least one hour left. Can you help Clemens again? a) Play a few games of Super Dragon game with your table neighbor to familiarize yourself with the Super Dragon game. b) First, investigate simpler variants of the Super Dragon game, where at the beginning there are 1, 2, 3, 4 or 5 pieces of lava on the table. c) Investigate simpler variants of the Super Dragon game, where at the beginning there are 6, 7, 8, 9 or 10 pieces of lava on the table. After the simple variants have been analyzed, things get serious again. d) Now investigate the real Super Dragon game with 24 pieces of lava.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_5

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Revenge: A New Game Against the Dragon

After the dragon has woken up, he plays a game of Super Dragon with Clemens. Clemens wins this game as well, much to the displeasure of the dragon: “Don’t show up here anymore!” Although Clemens has already successfully passed the adventure and won the longed-for magic ruby, here are two additional tasks to think about. e) Who can win the Super Dragon game with 41 pieces of lava? Is this the player who starts the game? f) What about the Super Dragon game with 43 pieces of lava? Anna, Bernd, Clemens and the Students Clemens defeated the dragon for the second time at play and won a pinch of dragon ointment. Anna and Bernd have learned that rule changes can have a big impact. This reminds of the subtask (a) from Chap. 3. They are proud that they were able to solve the fourth mathematical adventure quickly and safely. They were able to adapt what they learned in Chap. 4 well to the changed task. What I have Learned in this Chapter • A rule change can turn everything upside down. • Once again, I reduced a difficult task step by step to simpler tasks and solved them.

6

Word Puzzles and Graphs

On the outskirts of Right Angleton, there is a mysterious magic meadow. Very special flowers grow there, from which the bees collect their nectar to make magic honey. The bee Enigma lives in one of the beehives. Clemens is on his way to this magic meadow, where he wants to visit Enigma to get a comb with magic honey. Already from far away, he hears the buzzing of the bees. Enigma loves mathematical puzzles. And one thing is clear: Of course Clemens gets honey only if he can solve tricky mathematical puzzles. Today, Enigma sits on her favorite honeycomb and has nectar for exactly five cells. The honeycomb is shown in Fig. 6.1. Enigma wants to fill cells from left to right with nectar, one of each letter. In addition, the filled cells belonging to adjacent letters should have a common edge. The filled cells thus form a continuous chain of letters, resulting in the word “HONEY.” Enigma calls this a magical “HONEY” path. In order to be able to distinguish between the same letters, we write small numbers (indices) at the bottom right of the letters. In this way, we can clearly describe different “HONEY” paths. Figure 6.2 shows Enigma’s favorite honeycomb from Fig. 6.1 with distinguishable letters. a) Thus, H1 O1 N1 E1 Y1 denotes the path that leads through the topmost cells. Specify at least five more HONEY paths. b) In how many ways can Enigma fill up “EY,” if it has only nectar for two cells and starts with the letter E? Or to put it differently: How many (different) “EY” paths are there? c) How many “NE” paths are there if Enigma has only nectar for two cells and starts with the letter N? d) How many “NEY” paths are there if Enigma has only nectar for three cells and starts with the letter N? © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_6

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6 Word Puzzles and Graphs

Fig. 6.1 Enigma’s favorite honeycomb

H H H Fig. 6.2 Enigma’s favorite honeycomb with distinguishable letters

H1 H2 H3

O O

O

O1 O2 O3

N

N N

N1

N2 N3

E E

E

E1 E2 E3

Y Y Y

Y1 Y2 Y3

Clemens has found all paths consisting of two and three letters. But now he starts to think. “With five letters, it’s easy to miss a path, and the magic honey is gone,” Clemens thinks. Enigma notices that Clemens does not get very far. Although it is about her honey, Enigma helps him. “Do you remember what an undirected graph is, Clemens?” “Of course, Mercator Magicus explained it to me, and it helped me a lot in my first adventure. Otherwise, I certainly wouldn’t have gotten the wand. But I don’t see what use this is to me.” Enigma explains: “Look at Fig. 6.3, Clemens. There you see the same graph as in Fig. 2.2, but now there are arrows at the edges. By the way, mathematicians call this a directed graph,” Enigma explains. “In a directed graph, you can only pass the edges in the direction of the arrows. That’s similar to a one-way street.” Enigma gives Clemens the tip to present the problem more clearly. Actually, Enigma does not want to give honey at all, but she does not believe that Clemens can use her advice. She wants to appear generous.

6 Word Puzzles and Graphs

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Fig. 6.3 Directed graph (example)

e) Draw a directed graph where the letters (with indices) are the vertices. Connect two consecutive letters with a directed edge if the corresponding cells in the honeycomb touch each other. The arrow points from the preceding letter to the following letter. f) In how many ways can the “HONE” path H2 O2 N1 E1 be continued to a “HONEY” path? Find another “HONE” path that goes through E1 . How many ways can this path be continued to a “HONEY” path? g) How many ways can a “HONE” path ending in E2 be continued to a “HONEY” path? What does it look like when a “HONE” path ends in E3 ? h) How many “HONEY” paths are there? Enigma gives Clemens one last hint: “Use the results from subtasks (f) and (g) to simplify the task, and continue this strategy.” Clemens does not really know what to do now. Can you help Clemens? Anna, Bernd, Clemens and the Students Clemens has already successfully completed five mathematical adventures and acquired extremely useful magic utensils. Above all, Clemens does not have to serve the dragon for 99 years. And how are Anna and Bernd? They are very surprised when mathematics plays a role everywhere. They are confident that it will work out with the admission into the CoEYM.

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6 Word Puzzles and Graphs

What I have Learned in this Chapter • I now know what a directed graph is. • As with the games, another difficult problem has been simplified step by step, even though this time everything is actually quite different.

7

Even More Word Puzzles and Graphs

With your help, Clemens also passed his last mathematical adventure with flying colors. Enigma was very surprised that he managed it (she did not expect that) and gives him the promised honeycomb with the magic honey, a bit grumpy. She says: “You were very lucky! But I still have new tasks for you. If you can solve these tasks, you will also get three witch hazels.” “What can you actually do with witch hazel?” Clemens asks. “If you get stuck on a mathematical problem and throw a witch hazel firmly on the floor, you will get a tip. But remember: Each witch hazel can only be used once. After that, it is used up,” warns Enigma. “To make things more exciting: If you can’t solve the new tasks, you have to give me the honey back. Do you still want to try the tasks? But I won’t give you any more tips.” Clemens answers immediately: “Of course I’ll try!” a) How many possibilities are there to represent the word nectar in Fig. 7.1 as a continuous chain of letters (“NECTAR” paths)? Use what you learned in Chap. 6. First represent the honeycomb as a directed graph and then determine the number of “NECTAR” paths. b) How many possibilities are there to represent the word flower in Fig. 7.2 as a continuous chain of letters (“FLOWER” paths)? Proceed as in subtask (a). c) Think up a word puzzle for yourself and solve it. There should be at least 12 paths. Anna, Bernd, Clemens and the Students After the fifth mathematical adventure (Chap. 6) had taken him to his limits after all, Clemens mastered the sixth mathematical adventure relatively relaxed.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_7

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Even More Word Puzzles and Graphs

Fig. 7.1 Enigma supply honeycomb

N N

E E

Fig. 7.2 Another word puzzle

F

L

C C

O

T T

W

A A A

R R R R

O E L R W F O E L W R O

Anna and Bernd are also happy. “That went like clockwork,” says Bernd. And Anna adds: “We’ve already learned a lot.” Clemens has already completed six mathematical adventures and has acquired extremely useful magic utensils, namely a magic wand, a magic cloth, a magic ruby, a pinch of dragon ointment, magic honey and just three witch hazels. And what happened to Anna and Bernd’s wish to become a member of the CoEYM? The club chairman Carl Friedrich receives Anna and Bernd and says: “Anna and Bernd, so far you have done a great job! However, this is only half the story. To become a member of the CoEYM, you have to complete six more mathematical adventures with Clemens.”1 Anna and Bernd are visibly pleased with the praise from Carl Friedrich: “It has been incredibly fun and we have already learned a lot,” says Anna, and Bernd adds: “Together we have solved problems that we would certainly not have been able to solve alone. I never knew that mathematical statements had to be proven.” 1 This

refers to the mathematical adventures in Volume II.

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Even More Word Puzzles and Graphs

27

What I have Learned in this Chapter • I practiced the solution method from the last chapter again and understood it even better.

Part II Sample Solutions

Part II contains detailed sample solutions to the tasks in Part I. The target group are leaders of study groups for gifted primary school students, teachers and parents (but not the students). As a rule, this makes hardly any difference; only in some places is differentiation made. In order to avoid awkward wording, only the “instructor” is normally addressed in the following. Table II.1 shows the most important mathematical techniques used in the individual chapters. In the sample solutions, the mathematical goals of the individual chapters are also explained, and outlooks are given on where the mathematical techniques learned are still being used. It can give the children additional motivation and self-confidence when they learn that very advanced tasks can be solved with the techniques learned (see also the preface by Amann 2017). At the end of each task chapter, you will find a summary “What I have learned in this chapter.” This is a counterpart to Table II.1, but in a language suitable for students. The teacher can work out the learning progress together with the students. This can be done, for example, at the following course meeting to recapitulate the last chapter.

Table II.1 Overview: Mathematical contents of the task chapters Chapter

Mathematical techniques

Outlook

Chapter 2

Modeling of a real-world problem (path problem) as undirected graph, colored graph, mathematical proof

Coloring proof (Engel 1998)

Chapter 3

Analysis of the effect of small changes in the assumptions, coverage problems, mathematical proof

(Engel 1998), university pre-course

(continued)

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Part II: Sample Solutions

Table II.1 (continued) Chapter

Mathematical techniques

Outlook

Chapter 4

Mathematical game, optimal strategy with proof, tracing back to smaller problems

Game theory, mathematics competitions, historical: “electron brain” NIMROD against Ludwig Erhardt (Schmitz 2017)

Chapter 5

Mathematical game (see Chapter 4), optimal strategy with proof, trace back to smaller problems

See Chapter 10

Chapter 6

Modeling of a real-world problem (word puzzle) as directed graph, stepwise simplification of the initial problem

(Nolte 2006)

Chapter 7

Modeling of real-world problems (word puzzles), application of the techniques from Chapter 6

See Chapter 12

Main mathematical techniques and outlook

8

Sample Solution to Chapter 2

In the first mathematical adventure, there is no calculation. This is certainly a surprise for the students. Also new is the definition of an undirected graph and the realization that in mathematics, assertions have to be proven. First steps The instructor sketches the city map on the blackboard. The students show possible paths to the magic shop on the city map. a) Work with students to develop ways of writing down paths. For example, “l,” “r,” “u” and “d” for left, right, up and down or “w,” “e,” “n” and “s” for west, east, north and south. Work with students to define a spelling that will be used in the following. The sample solution uses “l, r, u, and d.” Didactic suggestion Divide the students for subtasks (b)–(e) into four groups. Each group works on one subtask. Give the students enough time to develop ideas together, discuss them in the group and present the solutions in the plenary. Didactic suggestion The division into four groups is tailored to SGs. In the context of differentiated teaching, it will probably not be possible to form more than two groups. In home schooling, one parent can take on subtasks, e.g., (c) and (d). b) c) d) e)

Possible paths: rrrdd, rdrdr, rrddr, drrdr, … Possible paths: Possible paths: rrrdudd, rrdlrdr, dddrurr … Possible paths: -

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_8

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Sample Solution to Chapter 2

Experience shows that groups 1 and 3 have found many correct solutions and proudly present them. Groups 2 and 4 have no solution or only wrong solutions, where the children miscounted the number of steps. After further unsuccessful attempts, the (correct) assumption arises in the children that subtasks (c) and (e) have no solutions at all. A “democratic vote” among the children will probably confirm this impressively, if they have only tried long enough unsuccessfully to find a solution. However, the matter is not so simple. Mathematical questions are not decided by majority decisions. Rather, we want to prove that subtasks (c) and (e) do not have solutions. It is an important goal of Chap. 2 that the students understand that in mathematics, assertions have to be proved and how to prove them. This, of course, goes far beyond the mathematics lessons in primary school. However, we are also dealing with primary school students who are particularly gifted in mathematics! As we will soon see, this proof can be understood by gifted primary school students. For this purpose, important intermediate steps were outlined in Chap. 2 and the definition of an undirected graph was introduced. Since the concept of a mathematical proof is fundamental on the one hand, but completely new to the students, the instructor should allow enough time at this point as long as the students make serious attempts to develop a proof themselves. f) On the way to our proof, the first question that arises is what information on the city map is ultimately superfluous to solve the problem. Since such information makes our task more difficult at best, we want to limit ourselves to the essential information. The children themselves will soon discover that the houses are superfluous. Without losing any information, we can make the city map a bit clearer (see Fig. 8.1). In Fig. 8.2, we go one step further. Clemens moves from intersection to intersection. Therefore, we paint a thick point on each intersection, which we connect with a line to the adjacent street centers. Clemens can get to all directly connected street centers in one step. g) With the abstraction of our problem, we have already made a good deal of progress. Let us continue along this path. Actually, the points representing the intersections and their connecting lines describe the complete situation. Just like the small houses, the street blocks can now also disappear without a problem. Figure 8.3 shows the city map of Right Angleton as an undirected graph, with vertices corresponding to the street centers and edges to the streets.

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Sample Solution to Chapter 2

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Clemens

magic shop Fig. 8.1 Simplified city map of Right Angleton (small section)

(What an undirected graph is, Magister Magicus explained in Chap. 2. We got to know directed graphs in Chap. 5, where the edges have directions). We can now formulate our question, mathematicians also like to speak of a “problem,” in this way: Is it possible to get from the starting point (Clemens’ house) to the end point (magic shop) in 6 or 8 steps in the graph in Fig. 8.3? We have already come a good deal closer to the solution. There is only one last step missing, which Mercator Magicus has already hinted at in Chap. 2. h) In Fig. 8.4, the vertices are colored black and white like a chessboard. (By the way, mathematicians speak of a colored graph.) Of course, other colors, e.g., blue and red, can be chosen on the board. In the sample solution, the colors result from the black and white print. Does the coloring of vertices bring new insights, except that it looks visually beautiful? You bet!

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Sample Solution to Chapter 2

Clemens

magic shop Fig. 8.2 Edited city map of Right Angleton (small section)

Observation When Clemens goes from one vertex to the next (in real life: from one street intersection to the next), the color changes. From a black vertex, he always gets to a white vertex and vice versa, from a white vertex to a black vertex. Clemens starts on a black vertex. After one step, he stands on a white vertex and after two steps he stands on a black vertex again. So it is clear: After an even number of steps (e.g., after 100 steps), Clemens always stands on a black vertex, after an odd number of steps on a white vertex. So: After 6 or 8 steps, Clemens stands on some black vertex. But the vertex in front of the magic shop is white! So Clemens cannot stand in front of the magic shop. So the tasks (c) and (e) really have no solutions. However, this insight now has a completely different meaning. We not only suspect it (due to many unsuccessful attempts to find such paths), but we have proven it. In order to save time, the first city map Fig. 2.1 can be converted step by step to Fig. 8.4 on a blackboard.

8

Sample Solution to Chapter 2

35

Clemens

magic shop

Fig. 8.3 Representation of the city map of Right Angleton as an undirected graph (small section)

By the way, already at the beginning of the proof, a student of the mathematics study group mentioned above essentially saw through the basic principle that Clemens cannot “win” a single step, but (understandably) could not provide a “rigorous” proof. Nevertheless a mature achievement for a primary school student! There are two subtasks left. But their solutions are now relatively simple. i) Similar to task (d), here Clemens at first can go 47 times each one step up and then down again. Then he stands on the starting point again, and he still has exactly 5 steps left to get to the magic shop. A possible solution is ud…ud(47 times)rrrdd. j) According to our extensive reflections on subtasks (c) and (e), this subtask is now really “child’s play”: the number 2,020 is even, and therefore there is no path of 2,020 steps.

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Sample Solution to Chapter 2

Clemens

magic shop

Fig. 8.4 Representation of the city map of Right Angleton as a colored graph (small section)

Mathematical Goals and Outlook For the children, this entry-level task is unusual for several reasons. First of all, they are amazed to discover that mathematics is not just about arithmetic. They probably see a mathematical proof for the first time in their lives. And above all, the children learn that mathematics is not just about applying “formulas,” but requires imagination and creativity. As a mathematical proof technique, the children get to know a colored graph, even though the pattern solution does not, of course, provide a systematic introduction to graph theory. They learn how to model a real-world problem mathematically in order to be able to solve it. Coloring proofs occur again and again in mathematics, for example, (as here) to show that certain facts are impossible. The collection of problems (Engel 1998) devotes an entire chapter to this topic (“Coloring Proofs”). In total, Engel (1998) contains about 1,300 tasks from more than 20 demanding national and international mathematics competitions (even from the International Mathematical Olympiad).

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Sample Solution to Chapter 2

37

The addressed readers are trainers and participants of competitions up to the highest level. For our purposes, however, the tasks in Engel (1998) are usually much too difficult.

9

Sample Solution to Chapter 3

After the very challenging tasks in Chap. 2, Chap. 3 continues much easier. The tasks in Chap. 3 offer the opportunity to apply what you have learned in Chap. 2 to simpler tasks. This is important to give the children self-confidence. a) In Fig. 3.1, an additional path is drawn, which diagonally connects two street centers. Does this change the solutions of the tasks in Chap. 2? Of course the solutions to the subtasks (b), (d) and (i) are solutions now, because no path was removed. But now there is also a path of 6 steps, e.g., rrr(dd)rd, where “dd” stands for “diagonally down.” Likewise, solutions with 8 or 2020 street sections can be found now; we only have to “tread on the spot” often enough, e.g., alternately go up and down. What happened to our beautiful proof? Of course you can create a colored graph like above, which describes the city map. But our proof does not work anymore: Observation While in the original city map from Chap. 2 you changed the color with every step, this is now mostly the case, but not always: If you walk the diagonal street, you get from a white vertice to a white vertice. With this our proof collapses. (And of course it must, because now there are paths with an even number of steps, as we have just found out). This is an example of how small causes can have great effects. Or to put it another way: In mathematics, details matter. b) This subtask should not be a problem. Figure 9.1 shows a possible solution. c) Subtask (c) has no solution, which is easy to understand. Each magic cloth covers exactly one white and one black square. With 31 magic cloths, 31 white and 31 black squares become invisible. So there must be one white and © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_9

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Sample Solution to Chapter 3

a b c d e

Fig. 9.1 A possible solution to subtask (b)

f

g h

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1 a b c d e

f

g h

one black square left in any case. But the squares a1 and h8 are both black. Therefore, no solution to subtask (c) can exist. Mathematical Goals and Outlook The students should practice the technique of coloring proofs once again in a different variation. Of course, you could also set task parts (b) and (c) for a monochrome board consisting of 8 × 8 squares. For the solution of subtask (c), there would then be an additional difficulty because the board would first have to be colored (e.g., black and white). Subtask (c) is well known in other contexts, partly in a slightly modified form. In Engel (1998), in the introduction of the second chapter (“Coloring Proofs”), it motivates the mathematical proof technique of coloring. By the way, a modified version of task part (b) (with other removed squares) was set as an exercise in the university preliminary course on formal methods of computer science for prospective computer science students of the Rhenish Friedrich Wilhelm University of Bonn (Rheinische Friedrich-Wilhelms Universität Bonn) in the winter semester 2014/2015.

Sample Solution to Chapter 4

10

In Chap. 4, a mathematical game is examined. However, the goal of a mathematical game is not to play light-heartedly, but to find the optimal game strategy. The students get to know this thought in the parts (b)–(e). a) Games are always interesting for children and therefore the task part (a) can be extended to about 15–20 min. Lego bricks or tokens are suitable as game pieces. Table 10.1 summarizes the results of subtasks (b) and (c). In the left column, the number of lava pieces with which the simplified version of the lava game is played is entered. The entry (G) means that this player can force the win if he plays the best strategy. b) It is obvious that for 1, 2 or 3 lava pieces the player in the turn can win. He only has to take away all the lava pieces on the table. Let us now look at the variant of the lava game that is played with 4 lava pieces. Player 1 can take away 1, 2 or 3 lava pieces. Then there are 3, 2 or 1 lava pieces left, but it is now the turn of player 2. Player 2 simply takes away all lava pieces that are still on the table. So we see that if you play a dragon game with 4 lava pieces, Clemens should not start the game at all! Observation Incidentally, the first three lines of Table 10.1 could also be used for this observation, since it is now player 2 s turn and the roles of the two players have thus been swapped as far as “it is his turn” is concerned. This may seem exaggerated at this point, but it is a useful observation! c) How does player 1 win when there are 5 pieces of lava on the table? With our preliminary considerations, this is now very easy to answer: He takes away © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_10

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42 Table 10.1 Dragon game: Who can force the win?

10

Sample Solution to Chapter 4

Lava pieces

Player 1 (next move)

1

(G)

2

(G)

3

(G)

4

(G)

5

(G)

6

(G)

7

(G)

8

Player 2

(G)

This table shows which player can force the win (G) when player 1 starts

exactly one lava piece! Then 4 lava pieces remain, and it is now the turn of player 2! But we already know that in a dragon game with 4 lava pieces, it is unfavorable to start the game. The same happens if there are 6 or 7 lava pieces on the table at the beginning of the game: Player 1 takes away 2 or 3 lava pieces in the first move. Next we look at the lava flow with 8 lava pieces. Table 10.1 says that player 2 can force the win, but how does that work? Player 1 takes away 1, 2 or 3 lava pieces, leaving 7, 6 or 5 lava pieces. But we already know that the player in turn loses if there are 4 pieces on the table. With this knowledge, player 2 simply takes away 3, 2 or 1 lava pieces, leaving exactly 4 pieces of lava. d) Observation Player 2 can force that he and player 1 together take away exactly 4 lava pieces (1 + 3, 2 + 2 or 1 + 3) in their first moves. In the variant with 8 lava pieces, player 2 transfers the game with 8 lava pieces into a variant with 4 lava pieces, where player 1 is in turn! We already know that this is won for player 2. How does this observation help us? A lot! It is the key to the solution of our task: Player 2 can step by step reduce the number of lava pieces by 4, and then it is the turn of player 1. For the lava game with 12 lava pieces this means: No matter what player 1 does, player 2 reduces the remaining lava pieces to 8 and then to 4 and finally to 0. e) And further: game variants with 4, 8, 12, 16, 20, 24, … pieces are always won for player 2, if he knows the right strategy. For the real dragon game with 24 pieces of lava this means: Clemens should not start in any case. If the dragon starts, Clemens reacts in such a way that 20, 16, 12, 8, 4 and finally 0 lava

10

Sample Solution to Chapter 4

43

pieces remain for the dragon. So Clemens can always win if the dragon makes the first move. Mathematical Goals and Outlook The lava play ends after finally many steps. In such games, it is often useful to analyze the game from the end. The decisive solution idea, namely that the second player can always force the two players to take together 4 pieces off the table in their next moves, was recognized by the analysis of small game examples with few pieces. The children get to know the technique of converting a difficult mathematical problem step by step into problems that are easier to solve; here the (original) dragon game with 24 pieces into dragon games with 20 pieces, with 16 pieces, …, and finally only with 4 pieces. As in Chaps. 2 and 3, the proof that the strategy described actually forces the win is clear and comprehensible and not vague or arbitrary (e.g.: “Player 2 can win because it has been so in several rounds of the game”). So the students have again given a strict mathematical proof. In this and the following adventure, two “take-away games” are examined in more detail. In the mathematical literature, you can find different variants of takeaway games with different degrees of difficulty. It is perhaps worth mentioning that during the Berlin Industrial Exhibition in 1951, the English “electron brain” Nimrod (original name; a mainframe computer) played and won three games of NIM (a more complicated take-away game) against the then Federal Minister of Economics and later German Chancellor Ludwig Erhard (Schmitz 2017). This adventure introduces the mathematical game theory. Game theory is a branch of mathematics that has numerous applications, including in economics. Unlike in the dragon game, a player cannot normally force the win, as random events influence the outcome of the game. The aim of these games is to determine game strategies that are optimal (in a sense to be specified). Mathematical games, or more precisely the search for optimal strategies, also play a role in mathematics competitions (see, e.g., Mathematik-Olympiaden e. V. 2009, Task 480941, 2013a, Task 520514, or various tasks from the Federal Mathematics Competition). Perhaps in the future, children will also think about optimal strategies for “normal” games.

Sample Solution to Chapter 5

11

This adventure continues the previous adventure. The rules of the dragon game are changed, and this will affect the optimal game strategy. a) Students should familiarize themselves with the changed rules and play against each other. Table 11.1 summarizes the results for subtasks (b) and (c). In the left column, the number of lava pieces with which the simplified version of the Super Dragon game is played is entered. The entry (G) means that this player can force the win if he plays the best strategy. b) For 1 lava piece, player 2 wins of course, because now the player who has to take the last lava piece from the table loses. In the Super Dragon game, it is obvious that the player who has the move can win for 2, 3, 4 or 5 pieces of lava. He only has to take away all but one of the lava pieces that are on the table. c) The variant of the Super Dragon game with 6 pieces is again favorable for player 2. No matter what player 1 does in his first turn: player 2 leaves exactly one gaming piece on the table, and player 1 loses. With 7, 8, 9 or 10 lava pieces, player 1 wins by removing 1, 2, 3 or 4 lava pieces from the table. Then 6 lava pieces remain, and as we have just seen, this is very unfavorable for the player whose turn it is. d) Observation Again, it is important that the player not in turn can always ensure that the other player and he remove exactly 5 pieces in their next two moves. (In the dragon game, the “critical number” was 4. It is important to

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_11

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46

11

Table 11.1 Super Dragon game: Who can force the win?

Lava pieces

Player 1 (next move)

1

Player 2 (G)

2

(G)

3

(G)

4

(G)

5

(G)

6

(G)

7

(G)

8

(G)

9

(G)

10

(G)

11

Sample Solution to Chapter 5

(G)

This table shows which player can force the win (G) when player 1 starts

work out this difference). Game variants with 1, 6, 11, 16, 21, … pieces are always unfavorable for the starting player. For the Super Dragon game with 24 lava pieces, this means Clemens should start. He wins by forcing with his moves the following intermediate scores: 21, 16, 11, 6, 1. e) In the game variant with 41 pieces of lava, player 2 wins by successively forcing the intermediate scores 36, 31, 26, 21, 16, 11, 6, 1. f) In the game variant with 43 lava pieces, player 1 can force the victory by taking away 2 lava pieces in the first turn, leaving 41 lava pieces. Now it is the turn of player 2, and we know from subtask (e) that with 41 lava pieces (if player 2 plays the best game), the player who starts loses. Mathematical Goals and Outlook This mathematical adventure is very closely related to the last mathematical adventure. The students thus have the opportunity to apply and deepen what they have learned.

Sample Solution to Chapter 6

12

In this chapter, students learn what a directed graph is. Subtasks (a)–(f) lead to (h). Similar to the Dragon and Super Dragon game, the initial problem is transformed into simpler problems in several steps. Tasks (a)–(c) are quite simple. They should be successfully completed by all students, which will give them their first sense of achievement. a) Examples of other HONEY paths: H2 O2 N2 E2 Y2 , H2 O3 N3 E3 Y2 , H1 O2 N1 E2 Y1 , H3 O3 N2 E3 Y3 , H1 O2 N2 E2 Y1 . b) There are 5 “EY” paths, and that is: E1 Y1 , E2 Y1 , E2 Y2 , E3 Y2 , E3 Y3 . c) There are 5 “NE” paths, and that is: N1 E1 , N1 E2 , N2 E2 , N2 E3 , N3 E3 . d) From this part on, this adventure becomes mathematically interesting. It is clear that any “NE” path can be supplemented to a “NEY” path. Due to the arrangement of the cells, there are one or two possibilities for this. We write down the “NEY” paths systematically. We keep the “NE” path order of task part (c) and complete the “NE” paths one after the other to “NEY” paths: N1 E1 Y1 , N1 E2 Y1 , N1 E2 Y2 , N2 E2 Y1 , N2 E2 Y2 , N2 E3 Y2 , N2 E3 Y3 , N3 E3 Y2 , N3 E3 Y3 That is a total of 9 “NEY” paths. Observation The “NE” path N1 E1 can only be continued to a “NEY” path in 1 way, namely with Y1 . The other 4 “NE” paths can be continued in 2 ways each. With this observation, we can easily calculate the number of “NEY” paths, namely there are 1 · 1 + 4 · 2 = 9 “NEY” paths, as we already know. This observation provides the key to the actual, more difficult task, namely to determine the number of “HONEY” paths without having to write them down laboriously one by one. Besides, it is also easy to forget a path.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_12

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Sample Solution to Chapter 6

Didactic Suggestion This observation should be discussed in detail with the students. As soon as these facts are clear, things can go on. The students should not be discouraged too quickly from looking for all “HONEY” paths. Systematically sorted (as in the sample solution to (d)), this can help to clarify the general systematics to the students. e) Figure 12.1 describes the honeycomb structure as a directed graph with the letters as vertices. For example, an arrow points from O2 to N1 because their cells have a common edge. A “HONEY” path is obtained by starting with an “H” and following the arrows. Subtasks (f) and (g) again take up the strategy from subtask (d), this time for complete “HONEY” paths instead of “NEY” paths. Since this is an important step for subtask (h), the students should practice this procedure again. f) The “HONE” path H2 O2 N1 E1 can only be supplemented to a “HONEY” path in one way, namely by Y1 . The same obviously applies to any other “HONE” path ending in E1 . Example: H1 O1 N1 E1 . g) As in subtask (f) the “history” “HON” is irrelevant. It is only important that the “HONE” path ends in E2 . Such a path can be continued by Y1 or Y2 , i.e., in two ways to a “HONEY” path. Likewise, any “HONE” path ending in E3 can be continued in two ways to a “HONEY” path. h) From here on, the task becomes really interesting, but also more difficult. The instructor should allow enough time for the last subtask. The task part (h) should be worked on together with the students. From (f) and (g), we already know that a “HONE” path ending in E1 can be continued in exactly one way to a “HONEY” path. If the “HONE” path ends in E2 or in E3 , there are two possibilities. Observation So if we knew how many “HONE” paths end in E1 (in E2 or E3 ), we would also know how many “HONEY” paths pass through E1 (through E2 or Fig. 12.1 Representation of the honeycomb “HONEY” as a directed graph

H1

O1

N1

E1

Y1

H2

O2

N2

E2

Y2

H3

O3

N3

E3

Y3

12

Sample Solution to Chapter 6

49

E3 ). More precisely: There are as many “HONE” paths that end in E1 as there are “HONEY” paths that go through E1 . There are twice as many “HONEY” paths that pass through E2 (or E3 ) as there are “HONE” paths that end in E2 (or E3 ). If one then added up the three numbers of “HONEY” paths, the subtask (h) would be solved. Unfortunately, we do not know yet how many “HONE” paths end in E1 , E2 and E3 , respectively, but this is an easier problem as there are no longer 5 but only 4 letters to consider. Figure 12.2 summarizes our findings so far. We have removed the last “layer” of vertices (representing the letters “Y1 ,” “Y2 ” and “Y3 ”) and the edges going there and thus we obtain a subgraph of the initial graph. The numbers in the square brackets after E1 , E2 and E3 indicate how many possibilities there are to continue a “HONE” path that ends at this point. These are referred to briefly as “remaining path numbers” in the figures. We will continue to pursue this strategy. A “HON” path that ends in N1 can be continued with E1 or E2 to a “HONE” path. The subgraph in Fig. 12.2 (or the honeycomb Fig. 6.2) shows that this “HONE” path in turn allows only one continuation (if it goes through E1 ) or exactly two continuations (if it goes through E2 ) to a “HONEY” path. This means that a “HON” path ending in N1 can be continued to a “HONEY” path in 1 + 2 = 3 ways. (Example: the “HON” path H2 O2 N1 can be added to the “HONEY” paths H2 O2 N1 E1 Y1 , H2 O2 N1 E2 Y1 and H2 O2 N1 E2 Y2 . However, we are not interested in the concrete paths, only in their number.) If the “HON” path ends in N2 , there are 2 + 2 = 4 possibilities; if it ends in N3 , there are only 2 possibilities, since E3 is forced. (For this, you add the numbers in the square brackets behind the E letters, which you can reach from the respective N.) So we have further simplified our task. Figure 12.3 illustrates Fig. 12.2 Word puzzle “HONEY”: Subgraph with remaining path numbers

Fig. 12.3 Word puzzle “HONEY”: Subgraph with remaining path numbers (2)

H1

O1

N1

E1 [1]

H2

O2

N2

E2 [2]

H3

O3

N3

E3 [2]

H1

O1

N1 [3]

H2

O2

N2 [4]

H3

O3

N3 [2]

50

12

Sample Solution to Chapter 6

the new situation. Figure 12.3 shows that each “HON” path ending in N2 can be continued in 4 ways to a “HONEY” path. If you turn the wheel back again, you get Figs. 12.4 and 12.5, but what does Fig. 12.5 say? The entry H1 [10], for example, means that each “H” path starting (and ending) in H1 can be completed in exactly 10 ways to a “HONEY” path starting in H1 . Corresponding statements apply to “H” paths that end in H2 or H3 . Of course, there is only one “H” path, which starts and ends in H1 (resp. in H2 , resp. in H3 ). Our continued reduction to ever shorter paths has thus borne fruit. But now task (h) is solved: In total there are 10 + 13 + 6 = 29 “HONEY” paths. Didactic Suggestion The solution procedure for subtask (h) is methodologically not very easy and can lead to difficulties especially for third graders. Depending on the composition of the SG and his previous experience, the instructor can omit subtask (h) or simply let the students search for all “HONEY” paths. The sample solution reveals how many paths there are. In this case, the instructor in Chapter 7 would have to formulate simpler subtasks, analogous to (a)–(g) in this chapter, or let the students search the “NECTAR” and “FLOWER” paths. Mathematical Goals and Outlook This chapter follows on from several earlier chapters in various respects. As in Chap. 2, a real world problem is first transformed into a graph problem (here: modeling as directed graph). Similar to Chaps. 4 and 5, a difficult initial question is systematically transformed step by step into simpler problems, from whose solutions one finally obtains the solution for the initial problem. Fig. 12.4 Word puzzle “HONEY”: Subgraph with remaining path numbers (3)

Fig. 12.5 Word puzzle “HONEY”: Subgraph with remaining path numbers (4)

H1

O1 [3]

H2

O2 [7]

H3

O3 [6] H1 [10] H2 [13]

H3 [6]

12

Sample Solution to Chapter 6

51

At this point, it should be noted that in Nolte (2006), various word and path puzzles are described. However, this essay does not go into detail about solution methods, but concentrates on didactic aspects and describes practical experiences.

Sample Solution to Chapter 7

13

The last mathematical adventure had been methodologically rather difficult and may have caused some frustration. In Chap. 7, we have to apply and deepen the solution method worked out in Chap. 6. So there are no particular difficulties lurking here. Nevertheless, this adventure is important to practice and deepen the solution method and to give the students further experiences of success. a) Figure 13.1 shows the honeycomb from Fig. 7.1 as a directed graph. In the next steps, we work from right to left as in Chap. 12 and transform step by step the initial problem into simpler problems (see also Figs. 12.2–12.5). Figure 13.2 describes the first step. We already know the further procedure from the last mathematical adventure in Chap. 6, so we only give the subgraphs with remaining path numbers (Figs. 13.2, 13.3, 13.4, 13.5 and 13.6) without further explanation. From Fig. 13.6, the solution finally follows: There are a total of 17 + 7 = 24 “NECTAR” paths. b) The solution of the second word puzzle “FLOWER” is analogous. First, we present the honeycomb again as a directed graph (see Fig. 13.7). The next steps are the same as for solving Chaps. 6 (h) and 7 (a). No further explanations are therefore given. Figures 13.8, 13.9, 13.10, 13.11 and 13.12 describe the solution for subtask (b).

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0_13

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54 Fig. 13.1 Representation of the honeycomb “NECTAR” as a directed graph

13

Sample Solution to Chapter 7

N1

E1

C1

T1

N2

E2

C2

T2

Fig. 13.2 Word puzzle “NECTAR”: Subgraph with remaining path numbers

R1

A2

R2

A3

R3 A1 [1]

N1

E1

C1

T1

N2

E2

C2

T2

N1

E1

C1

T1 [3]

N2

E2

C2

T2 [4]

N1

E1

C1 [3]

N2

E2

C2 [7]

Fig. 13.3 Word puzzle “NECTAR”: Subgraph with remaining path numbers (2)

Fig. 13.4 Word puzzle “NECTAR”: Subgraph with remaining path numbers (3) Fig. 13.5 Word puzzle “NECTAR”: Subgraph with remaining path numbers (4)

A2 [2] A3 [2]

N1

E1 [10]

N2

E2 [7]

N1 [17]

Fig. 13.6 Word puzzle “NECTAR”: Subgraph with remaining path numbers (5) Fig. 13.7 Representation of the honeycomb “FLOWER” as a directed graph

A1

N2 [7]

F1

F2

L1 L2 L3

O1 O2 O3 O4

W1 W2 W3

E1

E2

R1 R2 R3

13

Sample Solution to Chapter 7

Fig. 13.8 Word puzzle “FLOWER”: Subgraph with remaining path numbers

55

F1

F2

Fig. 13.9 Word puzzle “FLOWER”: Subgraph with remaining path numbers (2)

L1 L2 L3

F1

F2

Fig. 13.10 Word puzzle “FLOWER”: Subgraph with remaining path numbers (3)

O1 O2 O3 O4

L1 L2 L3

F1

F2

Fig. 13.11 Word puzzle “FLOWER”: Subgraph with remaining path numbers (4)

W1 W2 W3

O1 O2 O3 O4

L1 L2 L3

F1

F2

E1 [2]

E2 [2]

W1 [2] W2 [4] W3 [2]

O1 [2] O2 [6] O3 [6] O4 [2] L1 [8] L2 [12] L3 [8]

Finally, it follows from Fig. 13.12 that there are 20 + 20 = 40 different “FLOWER” paths. Fig. 13.12 Word puzzle “FLOWER”: Subgraph with remaining path numbers (5)

F1 [20]

F2 [20]

56

13

Sample Solution to Chapter 7

c) A sample solution for task part (c) cannot, of course, be given here, as the students think up the tasks themselves. But the solution method should be clear. Didactic suggestion The instructor should give several students the opportunity to present their honeycomb and their solution on the blackboard. This will practice presenting their own solutions and all students will have the opportunity to re-enact the general solution method.

Mathematical Goals and Outlook See Chap. 12.

What You Learned From This essential

This book provides carefully designed learning units with detailed sample solutions for a mathematics SG for gifted students in primary school. In six mathematical stories, you have • • • •

learned how to model and solve real-world problems as graph problems. reduced difficult problems step by step to simpler ones. analyzed simple games and determined the optimal game strategies. learned that proofs are necessary in mathematics, and you have provided proofs yourself in different contexts.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 S. Schindler-Tschirner and W. Schindler, Mathematical Stories I – Graphs, Games and Proofs, Springer essentials, https://doi.org/10.1007/978-3-658-32733-0

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References

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