Prime numbers have continued to draw the attention of mathematicians for more than two millennia. In this Appendix, we present a brief overview of the type of questions that were raised and have been attempted to be answered. In the process, we mention some historically important theorems and conjectures concerning prime numbers. Proofs requiring advanced topics in mathematics have been omitted. Progress made so far in relation to yet unproven conjectures has also been discussed. Some interesting anecdotes and recreational mathematics in the history of prime numbers have been included. It may be mentioned that the use of prime numbers in the modern day computer cryptography and cryptanalysis, i.e., coding and decoding of secret information exchange, has already been discussed in Appendix II.

1. Introduction

For more than tow millennia, prime numbers have fascinated the greatest of mathematicians of every era. As we all know, prime numbers are those number, which have only trivial factors 1 and the numbers itself. The fundamental theorem of arithmetic says that all non-prime or composite numbers can be factorised in terms of prime numbers in a unique way (but for the order), as exemplified below:

18 = 2 × 3 × 3 = 3 × 2 × 3, etc.

Let us start with a list of some primes up to 1000:

2, 3, 5, 7, …, 23, 29, 31, 37, …, 769, 773, 787, 797, …, 821, 823, 827, 829, …, 977, 983, 991, 997

From the above list, it may be noticed that 2 is the only even prime, all other primes are odd. After 1859, mathematicians agreed to exclude 1 from the list of primes.

This is not a textbook on school mathematics, but a collection of somewhat uncommon problems in elementary arithmetic, algebra, geometry and probability theory. Most of the problems in the text are grouped under various categories, as mentioned in the ‘Contents’. A large majority of the problems is of ‘Puzzle’ variety, i.e., clever application of elementary knowledge of mathematics and common sense logic, rather than any tools of advanced mathematics, are required to solve these problems. The arrangement of the problems does not follow any graded pattern; so any problem can be attempted without solving the previous ones. Each problem is followed by asterisk mark(s), signifying the level of difficulty, viz., easy (*), moderate (**) and tough (***).

The section ‘Historically Famous Problems’ containing 25 problems, hopefully, gives a glimpse into how the great mathematical minds work. Some very special theorems and results in elementary mathematics (normally not mentioned in school textbooks), which can be regarded as mathematical gems and curiosities, are also included in this section. The difficulty levels of these problems have not been identified.

Detailed solutions, sometimes with alternative solutions, are included for all the problems.

The list of problems and their solutions is followed by four appendices. Appendix I contains the general methodology of solving linear Diophantine equations in two variables. This topic is not covered in high school mathematics, but solutions to a number of puzzles finally require solving such equations.

For secret message transfer through computer, every message is encrypted following a scrambling rule and the coded message is decrypted if the scrambling rule (i.e., the encryption key) is known. For a computer, the message (M), the key (K) and the coded message (C) are all numbers, expressed as strings of 0's and 1's. The one-way modulo function (which enables us to determine uniquely the RHS form the LHS, but not the other way round) and the difficulty of the obtaining two large prime factors (p and q) of a number N = p × q are combined to create an asymmetric key. Different keys are used for encryption and decryption.

Public Key and Private Key

Everybody knows my public key (like my telephone number). Anyone can send me an encrypted message using my public key, but only I can decipher it by using my private key.

RSA, or Rivest, Shamir, Adleman, algorithm, reported in 1976. depends on the difficulty of getting back two very large prime numbers p and q from their product N.

N (known to all) serves as the public key and p, q (known only to me) comprise my private key.

Martin Gardner (1977) asked the readers of Scientific American to find the two prime factors of a large number consisting of 129 digits.

This book is a collection of mathematics problems from various sources, which are listed as references. Most of the problems (and some solutions) are followed by a number within [], which indicates the reference number of the source. It may be pointed out that some problems can be located in more than one source. For some problems, no source has been mentioned. Either these problems are too common to mention a particular source or I received from some friends and the original source is not known to me. Now I must say ‘sorry’ to the authors of the original sources and this lapse is totally unintentional. Undoubtedly, my biggest debt is to the king of puzzles of the modern era, Martin Gardner. Next in line are Professors Ian Stewart and Ross Honsberger, who authored various books with excellent collections of problems to excite both non-mathematician adults and young students.

This collection of problems is not intended for teaching mathematics and, therefore, not much effort has been put to make a systematic or graded presentation. It is aimed at high school students and those who have some interest in elementary mathematics.

I am thankful to a number of my friends who encouraged me to put together this collection in the form of a book. Special thanks are due to Dr Raminder Singh, who went through almost all the problems and the solutions and suggested improvements.

IN recent decades, peace and conflict studies have begun to take roots as a significant interdisciplinary site of teaching and research. It has proliferated globally in its various rubric ranging from peace research, peace studies, conflict resolution, conflict management and conflict transformation among others. In South Asia too, the academic programmes that teach and research on peace are rapidly gaining a foothold. Increasing number of universities and colleges in the region have set up independent departments and Centres dedicated to peace studies. It is a befitting recognition to the perennially growing template enriched by scholars and philosophers across cultures.

However, the fledgling field of peace and conflict studies is beset with a range of contestations and criticisms. There are issues about its overwhelming and often schematized research agenda, the domination of positivists and heavily quantified methodology, uncritical stance towards terminology and the disconnection between research and action. One of the lingering challenges of teaching or researching peace and conflict in South Asia has been the enduring shadow of western perspectives, which tend to relegate indigenous discourses and frameworks. In fact, most of the pedagogic approaches in the region have imitated, evolved and intersected around western perspectives often reflecting their conceptual trajectories as well as faultlines (Upadhyaya 2013). There are, of course, some notable countercurrents and alternative perspectives in South Asian scholarship, which have critiqued the western-inspired notions of peace, often with a post-colonial bent.

Introduction

South Asian discourses on peace can draw much from Gandhian visions. His comprehensive approach to peace reflected in his oft-cited statement, ‘Earth provides enough to satisfy every man's need, but not every man's greed’, reveals a unique synergy between peace and sustainable development. Peace for Gandhi was not just the absence of war or violence, but also a pathway to expand human potential without hurting others in the community or in the larger ecosystem. No wonder Gandhi continues to influence the global thought process despite political and cultural transformations. Numerous thinkers and activists across the world swear by Gandhi's philosophy, and no global dialogue on peace can progress without reference to his ideas. Amidst the current global crises marked by a rising scale of violence, a worsening ecological crisis, terrorism and discriminatory patterns of globalization, Gandhian expressions on peace and sustainability assume critical salience.

Gandhi envisioned a holistic yet critical vision of peace that rejected violence in its entirety. He not only abhorred wars and killing under any guise, but also addressed the insidious ramifications of ‘structural violence’, embedded in structures and cultures. His disavowal of violence amidst the gravest of provocations makes him the most inveterate proponent of nonviolent methods to achieved peace. He firmly believed that positive peace could be brought about only by peaceful means and never by non-peaceful means (Galtung 1996). He believed that the modern state based on force is incapable of dealing with the forces of disorder, whether external or internal. Gandhi’s vision has inspired multiple streams of pedagogy and new thinking around such areas as sustainable development, human security and structural violence.

Peace and conflict studies are proliferating across disciplines in growing number of institutions and locations in South Asia, like elsewhere in the world. New interdisciplinary centres and departments have emerged which teach and research peace and conflict studies in its varied rubrics. This is in addition to the rising numbers of research institutes and think tanks dedicated to conflict analysis and peacebuilding. The subject matter of peace and conflict traditionally a preserve of international relations is drawing ever more on other disciplines like political science, sociology, psychology, anthropology, philosophy, religious and cultural studies and many others. Its scope is further widened by the inclusion of human security and non-traditional security concerns in the peacebuilding agenda. This is indeed a welcome development. The complex matrix of peace and conflict issues especially in South Asia certainly warrants deeper probe on a much wider template.

However, the budding field of peace and conflict resolution in South Asia is facing a crisis of relevance. Like in other areas of social sciences, it is caused by the unbridled sway of western pedagogies. With the exception of an uncritical delineation of Gandhian vision, there are not many indigenous approaches and critical insights, which inform the curriculum or methodology in this area. Notwithstanding a few counter currents posed by post-colonial and critical peace scholarship, the existing field of peace and conflict resolution is taught and disseminated in the region around western perspectives. This is not as much due to the paucity of knowledge and research on the subject, but more to the fact that the epistemological constitution of the disciplinary parameters that hardly allows voices from the marginalized southern hemisphere. The positivist and secular sway in the realm of conflict resolution, for instance, would not reckon with religion-based ideas and actors. The South Asian academic institutions too lag behind their northern counterparts in drawing indigenous pedagogies and analytical tools. All this has led to a significant epistemological and discursive deficit inpedagogies and research framework of peace and conflict in South Asia.

Introduction

This chapter outlines the long quest for peace and justice in Pakistan. It builds on the successful Lawyers' Movement launched in March 2007. This civil resistance successfully reinstated, in March 2009, the Chief Justice of Pakistan, Iftikhar Muhammad Chaudhry, who had been forcibly removed from office by General Parvez Musharraf in 2007. The movement was a remarkable instance of peaceful civil resistance against an entrenched military regime. The lawyers gave astonishing support to Chief Justice Chaudhry, who used the law to promote justice, in the years during Musharraf's military dictatorship. This ‘Black Revolution’ engaged people across gender, racial, ethnic and ideological affiliations throughout the country. The movement lasted two years, eventually achieving its goal of re-instating Chaudhry as Chief Justice and bringing down the dictatorship. It does however leave unanswered questions on whether these types of movements always promote peace and democracy in a society when they are heavily laden with their own professional interests.

Violence has been one of the basic expressions of Pakistan's politics since its emergence amidst a bloody sub-continental partition. Its protracted conflicts with India, along with the surge of Islamic fundamentalism and sectarian conflicts, served as fuel for violent insurgent elements. The democratic impulse remained overshadowed by the important role of the military in governance. Pakistan’s multifaceted involvement in Afghanistan’s upheavals and its uneasy nexus with the American alliance, first as the ‘frontline’ state against Soviet occupation, and more recently against Taliban forces, has only exacerbated the level of coercion and violence in its polity. There was indeed limited scope of non-violent resistance amidst such violent upheavals and political instability.

Introduction

Does the regime established on the basis of popular movement always contribute toward peacebuilding and strengthen the democratization process? If it does, what are the necessary ingredients for that? This chapter deals with the case of Nepal where frequent regime changes, by using so called popular movements, have paralysed the country. It appears that a practice has been established wherein every comfort, discomfort, approval, disapproval or breach of law either by the state or non-state agencies are being challenged through severe street protests. Many times these acts stand antinomies to democracy. Rise of various interest groups and non-state-actors, on the other hand, have further aggravated problems as their activities are maintaining permanent nature of revolutionary political culture with strategy of what Gramsci had called the movement of war manoeuvered. Professional political elites (political entrepreneurs), for their part, are found to have been exploiting state and its agencies in the name of democracy and peace. Such construction in political and social behaviour poses threat to political stability.

This chapter discusses the post-2006 politics and argues that regime change in Nepal has failed to work as per the spirit of popular movement which envisaged peaceful, prosperous, and politically stable Nepal. Ongoing political process, in contrast, is occupied by power politics where the agendas of ‘people’ at large are rarely discussed. Democracy, whose Nepali equivalent translation has now become loktantra (which used to be known by the name of prajatantra in the past) is reinforcing neo-patrimonial culture.

Introduction

This chapter seeks to focus on one of the action areas of culture of peace – fostering a culture of peace through education, and in doing so it engages with Gandhian thought and practice. It seeks to answer the following critical questions: How do we build a ‘Culture of Peace’ through education for peace? What is the state of affairs of mainstream school education in India? Is the school education in India geared towards building a culture of peace? Do Gandhian ideas and perspectives on education find a place in the Indian education system? How can Gandhian perspectives on education contribute to the building of a culture of peace?

The All-pervading ‘Culture of Violence’

The world has already been witness to two World Wars. The present generation has also seen the horrors of 9/11 from close quarters. The United Nations was founded on 24 October, 1945 with the sole aim of preventing another world war and this was clearly delineated in the Preamble of the Charter of the United Nations, which stated, ‘We the peoples of United Nations (are) determined to save succeeding generations from the scourge of war, which twice in our lifetime has brought untold sorrow to mankind’. In spite of this clear determination shown at the time of establishing the United Nations, there have been numerous small and big violent wars resulting in huge losses of life and property and insurmountable misery. Besides, we have also witnessed and continue to see around us, different kinds of violent conflicts– inter-state, intra-state, ideological, ethnic and communal, factional, conflicts over resources etc. at various levels – inter-personal, group and community level, social, national and international.