Books I read in 2013

I was fortunate to read quite a few good books in 2013. Here is the list of them except for those that I couldn’t finish or which I thoroughly disliked. Initially I wanted to write just a one-sentence blurb about each, but I couldn’t resist expanding them into short summaries/reviews in some cases. (Here was my 2012 reading list.)

If you have read any of these books and came away with an opinion totally different from mine, I would love to hear about it.

Part 1

Steven Callahan: Adrift: Seventy-six Days Lost at Sea. This short book tells the story of Steve Callahan, who suffered a sailing accident a few hundred miles off the coast of the Canary Islands, then floated on an inflatable liferaft for 76 days, until reaching Guadeloupe on the other side of the Atlantic, together with the small ecosystem that developed around the raft. The book is fascinating, both the descriptions of concrete events (the obtaining of water and food for survival, maintenance of the various systems on the boat) and the philosophical commentary about hope and despair, solitude, man’s place in nature, etc. are amazingly well-written. One of the best books I read in 2013.

Eric Weiner: The Geography of Bliss. The admittedly light-hearted question that the book explores is what makes people in different parts of the world happy – the answer could have been simply “it depends on the person” – but what we get instead is a travelogue of sorts through a number of faraway places, where the author meets and interviews people, sharing with us his own observations intermixed with “a new study has found…”-type of pop science and with tired stereotypes about each place. I wanted to like the book, the text has a nice flow and the author does make some amusing and occasionally some insightful observations – but for the most part I found it shallow and didactic, lacking any real curiosity or reflection, the author trying too hard in some parts to sound cute, in other places writing in an annoyingly defensive style – overall, a somewhat entertaining read, but quite pointless, unfortunately.

Albert Hofmann: LSD: My Problem Child. This is a leftover from 2012 which I forgot to write about last time. The book is a short autobiography of the chemist who inadvertently invented LSD – an interesting story well told, and an exceptionally clear and objective look at the effects of the invention, its potential, its dangers, and the ramifications to society. We meet interesting characters, including Timothy Leary, Ernst Jünger, and Aldous Huxley, and we follow the author and others on trips into to the mountains of Central America where they go through various psychedelic experiences.

Richard P. Feynman: The Meaning of It All. The book contains the transcripts of three public lectures given by Feynman in 1963 for a non-specialist audience. In his usual entertaining, anecdotal style, he discusses many loosely related topics, such as the scientific method, the value of science to society, religion, pseudoscience, dictatorships, conspiracy theories, and so on. Overall, a nice and accessible read, but less deep and original than his “Surely you are joking..” book, and apparently published against Feynman’s stated intentions.

Part 2

Karen Pryor: Don’t Shoot the Dog! The book talks about techniques for getting dogs, cats, dolphins, and other animals to do (or not do) various things. The methods are gentle, they are mostly based on well timed reinforcement signals and rewards, and the research and philosophy behind them seems sound. As the author points out through many examples, some of the techniques are no doubt useful when trying to influence human behaviour too (our own or of others) – for example when teaching children, or when trying to (un)learn some habit. There are also insightful observations about certain behaviors and phenomena, such as e.g. superstitions – overall, however, I found the method slightly oversold – and some of the case-studies involving human situations seemed a bit manipulative.

Maria Tatar: The Hard Facts of the Grimms’ Fairy Tales. After recently re-reading some of Grimm’s fairytales in original, I was curious to read Tatar’s analysis, as she appears to be a leading expert on the work. While it is interesting, the book also disappoints slightly: by “hard facts” she really means just that – we learn a bit (too much) about the biography of the brothers, the author dispels many myths about the way the tales were collected and edited, and she gives a detailed catalog of all the types of characters, heroes, villains, beasts, magical items, as well as the different turns of plot appearing in the tales. Few comments are made on deeper psychological symbolism of the tales, when such interpretations are mentioned, it is usually only to ridicule them. In large parts of the book, the author convincingly proves what the tales aren’t, without giving alternatives as to what they are instead. All in all, I found the book to be a well researched scholarly work, but (a bit) less colourful than expected.

Nelson Mandela: Long Walk to Freedom. The autobiography of Nelson Mandela, his personal life and the anti-Apartheid struggle: detailed, inspiring, written in an understated, no-nonsense tone and with exceptional clarity, radiating with optimism and strength on every page – one of the best non-fiction books I read in 2013.

Antal Szerb: Journey by Moonlight. I cheated here and I read this in the original Hungarian, but according to reviews, the English translation isn’t too shabby either. This is one of just about three or four novels written by Antal Szerb, who died tragically in 1945 and who was also an exceptionally erudite scholar and historian of literature. From such an author one might expect a “scientifically” written novel, but contrary to expectations, the novel is accessible and playful – it has subtle twists of plot, it shows deep understanding of character, and it never seems to take itself too seriously. This was by far the best fiction I have read in 2013.

Part 3

John Maynard Keynes: The Economic Consequences of the Peace. It is a rare opportunity to have one of the decisive historical moments witnessed and narrated by one of its sharpest and most eloquent contemporaries. In this case, this is exactly what happens, as Keynes gives a detailed account of the Paris Peace Treaty, an opinionated description that has more or less become the canonical interpretation of the historical event. He mixes keen observations on the characters (and character flaws) of the main participants, Clemenceau, Lloyd George, Wilson, and others, with a detailed analysis of the economic contents of the treaty and its projected consequences. The description is interesting for the back-of-the-envelope estimates Keynes makes, and for the detailed snapshot of how the world economy looked around the time of the first world war. The predictions about the untenability of the treaty and the ensuing catastrophes have mostly stood the test of time, further cementing Keynes’ reputation.

Richard Brautigan: Trout Fishing in America. An amusing short novel written with dry humour, containing small vignettes about various characters in rural (and occasionally in urban) America. One source of the humour is the usage of “trout fishing in America” in several ways: in its original meaning referring to the act itself, as the name of a person, as an adjective, etc.. Some reviewers compare Brautigan to Hemingway – which seems accurate – but the sensation one gets from this small book is that of an angrier, less focused but sometimes more imaginative author.

Richard P. Feynman: Perfectly Reasonable Deviations from the Beaten Track. This is a selection of Feynman’s personal correspondence, lovingly compiled by his daughter Michelle. We can follow through Feynman’s career and personal life, learn about his working style, travels, everyday routine, opinions on science, education, politics, etc. Some of the letters contain advice to colleagues or strangers, others are replies to fans, dispelling misunderstandings in physics, with patient and crystal clear explanations, some are diary-like accounts of events, others concern official business. Especially funny are the letters that show Feynman’s ambivalence towards accolades such as honorary degrees, academy membership and the Nobel prize. Also interesting are Feynman’s big picture explanations of theoretical physics and science in general, as well as his reviews of elementary textbooks – particularly his poignant criticism of the “New Math” of the 60s. The text is obviously less polished than his work written for a general audience, but nevertheless, it gives a glimpse into Feynman’s amazing (and amusing) personality, sharp mind and intellectual honesty, and occasional goofiness. The collection is worth a read even just as an example of a no-nonsense, precise style of letter-writing.

E. F. Schumacher: Small Is Beautiful. Schumacher’s 1973 book was ahead of its time in many ways, bringing up topics which today are known as sustainable development, environmentalism, anti-consumerism, simple living, organic movement, “go local” movement, regional development, decentralisation, etc. Some of the predictions in the book proved to be slightly inaccurate, and some of the statements might be overly general, but these detract only marginally from the overall message, which seems as relevant as ever and at a large scale as ignored as ever. The author’s main point is a criticism of global economy which he sees to be on a collision course and of prevalent economic philosophies that can be summed up as a pursuit of growth at all costs and an “idolatry of gigantism”. In Schumacher’s words “the most striking thing about modern industry is that it requires so much and accomplishes so little”. He sees as illusory the desire to create, in Gandhi’s words “systems so perfect that no-one will need to be good”. He exposes many inherent contradictions of mainstream economics and proposes sensible (if sometimes naive) alternatives. Memorable concepts of the book include: small scale operations, meta-economics, Buddhist economics, appropriate technology, intermediate technology. An inspiring and timely short book.

Part 4

Imre Lakatos: Proofs and Refutations. This acclaimed book contains a series of Socratic dialogues on the nature of mathematical discovery and proof, using Euler’s polyhedron theorem as a running example. The dialogues explore how definitions and theorems co-evolve, the role of formal and informal arguments, and most importantly, of local and global counterexamples. The dialogue format makes the book a pleasant read and also allows the author to present differing historical and modern views of mathematics.

Jean-Dominique Bauby: The Diving Bell and the Butterfly. A short memoir whose text was composed by selecting letters one-by-one, the author only being able to use his eyelids, as he suffered from locked-in-syndrome caused by a massive stroke, and died shortly after finishing the book. Some of the chapters depict small episodes from his earlier life, others explain his situation after the accident and his strategies for dealing with it. Tragic, hopeful, poetic, occasionally humorous, humbling. One of the most powerful books I’ve read.

Keith Johnstone: Impro: Improvisation and the Theatre. Impro seems to be the classic book on improvisation theatre, but it is a very enjoyable read for someone with little knowledge (and perhaps interest) in theatre too. Firstly it offers a glimpse into the mechanisms of a strange but beautiful microcosm (that of improv theatre) – it is always fun to learn about the concerns, the “rules of the game”, the system of values of some subculture totally different from our own. Furthermore, any field of human activity has its own metaphors, it influences thinking in its own way, and it spills over its boundaries, giving a toolset for its practitioner to tackle problems in other domains of life – thus taking a shot at becoming a “theory of everything”. This book is improv’s attempt at doing so, and not an unsuccessful one at that. We find insights about improvisation, acting, status, spontaneity, creativity, storytelling, concepts that are relevant to any human interaction, and the author has interesting things to say about all of them. The last part is a bit different, as the author discusses his experiences about teaching improvisation with masks, and relates it to trance-like states and hallucinations – slightly bizarre, but powerful and interesting nevertheless.

Werner Herzog: Conquest of the Useless. Herzog’s very entertaining diary from the time of production of the movie Fitzcarraldo. Mirroring the plot of the movie itself, the production is an endless ordeal in the Peruvian rainforest, apparently dragging on from failure to failure until the eventual completion. Besides the forces of nature and the bureaucratic obstacles, the erratic behavior of lead actor Kinski sets the tone for the book. Herzog gives a detailed account of the over-the-top everyday events of the production, as well as his own equally over-the-top commentary and reflections.

Part 5

Douglas R. Hofstadter: Fluid Concepts And Creative Analogies. In this book Hofstadter describes the research projects of his group, on attempting to model human thinking and problem solving. Rather than looking at the full messiness of the real world, the projects concern abstract microworlds, such as arithmetic puzzles of a certain kind or anagrams of words. Some of the studied problems are interesting in their own right – starting from a simple puzzle in recreational mathematics, deeper concepts are reached, others are thought-provoking due to their open-endedness, such as the analogies between strings in the CopyCat project. The approaches appear original and unconventional compared to mainstream AI or machine learning research – although they mostly fall short of efficiently solving the given problems. Hofstadter insists that his main goal was not efficiency (he doesn’t have much praise for world champion chess programs) – but rather to approach the problems in psychologically realistic ways. This might explain why there is no mention of computational complexity or other central concepts of computer science – however, without this connection, the study of heuristics seems a bit naive. This shortcoming is less apparent in the domains where the objective itself is ill-defined – for example when looking for a “reasonable” or “nice” continuation of a sequence of numbers. Here I think that it would have been beneficial to relate to other classical concepts, such as Kolmogorov complexity.

The most interesting parts of the book are those in which the author speculates about the workings of the mind, analyzing the way humans might approach and solve certain puzzles. Hofstadter has a knack for introspection, deconstructing a thought process into small explicit steps that seem entirely plausible on reading but which would be performed mostly unconsciously. These parts have a similar feel to Pólya’s “worked problems” in his books about problem solving. However, when the insights are translated into programs something is lost – while Hofstadter sees intelligence as an emergent phenomenon, the programs still appear to have his high level ideas for the particular domain “baked in” – something he vigorously criticizes in other projects. Taking all this into account, the results produced by the programs are rarely surprising.

The book is accessible and slightly more focused in scope than other books of Hofstadter, although there are some loosely related digressions. Some chapters are devoted to criticism of other research projects of the time – these seem to be mostly of historical value only. The “letter spirit” concept in the last chapter is fascinating on its own, but again, it tells more about the creativity of Hofstadter and his colleagues than about creativity or intelligence in general.

Seymour A. Papert: Mindstorms: Children, Computers, And Powerful Ideas. The book describes Papert’s vision of learning and the profound change that computers can bring about in education, especially at an early age. Inspired by Piaget’s theories and starting from his own experience of “constructively” learning mathematical concepts while playing with gears as a child, Papert presents a set of loosely connected ideas about curriculum-free, explorative learning and creativity, particularly related to mathematics. One of these ideas is to avoid a complete separation of the abstract thinking from physical, sensory-motor experiences. Another idea is that of a participatory, spontaneous, almost unconscious learning. As a concrete embodiment of these concepts, Papert advocates the use of his Logo system (turtle geometry) as an instrument for exploration. Such a use of computers by children is markedly different both from playing videogames and from using educational software that are little more than glorified textbooks: Logo lets children discover geometry in a natural, “local” way, it also effortlessly teaches modular, procedural thinking, and other algorithmic concepts. Papert’s vision is compelling, and a certain aspect of it has become reality: computers have indeed become powerful and widely available to children. However, the sweeping transformation in education that Papert envisioned has not been realized. While there are many options available to individuals, the mainstream trend has hardly been towards experimentation and imaginative tinkering, and more towards passive consumption and endless distractions. The “angry birds” of today seem to have far less potential than the turtle geometry of the 60s. Having already been fond of Logo, I still learned a great amount from the book, and I found it both informative and inspiring.

Marc Kac, Gian-Carlo Rota, Jacob T. Schwartz: Discrete Thoughts: Essays on Mathematics, Science and Philosophy. A beautiful collection of essays on the nature of mathematics and the state of its various subfields. A recurring theme is the interplay between experimental mathematics, applied fields, and the mathematical problems they inspire on one hand, and abstract, internal development of mathematics on the other – unsurprisingly, the debate on the relative merits of the two sides goes back to the ancient Greeks. The book is indeed less indiscreet than its companion, the quality of the writing and the depth of the essays is slightly uneven, but most parts of the book are interesting and stimulating.

Richard Branson: Losing My Virginity. The autobiography of Virgin Group founder Richard Branson – the story of his childhood and youth, his first business ventures, the early days of Virgin Music and Virgin Records, mainstream success after discovering Mike Oldfield: the ups and downs of building his business empire, personal life events, his sailing, ballooning and other stunts, the different domains into which his group expands and his philantropic initiatives. Overall, very entertaining and well written, interesting to learn about his approach to business, and the particularities of different businesses, as well as about his everyday life – a book filled with optimism and energy.

Part 6

Nassim Nicholas Taleb: Antifragile. After having thoroughly enjoyed Fooled by Randomness and Black Swan, I was eagerly awaiting the latest installment from Taleb – this time on the concept that is the opposite of fragility, which is, as he insists, not just robustness, i.e. resistance to damage, but antifragility, i.e. the capacity to actually improve in the face of mild stress, or of random perturbations. Taleb explores this deceptively simple concept in depth, bringing examples from economics, biology, medicine, ethics, and everything in between.

When it comes to Taleb’s three main books, many (though not all) of his critics admit the sharpness and originality of his ideas. Many are put off, however, by the strongly stated opinions and the aggressive, often self-congratulatory tone. On the contrary, I actually found these traits contributing to the charm of the book, it is enjoyable to read how commonly held beliefs are plain wrong and how everyone is stupid (as long as we imagine a “those present excluded” fineprint while reading, of course) – however, I was slightly annoyed by some of the repetition, and some of the case studies which seem more shaky than the others. The book gets much better towards the second half – Taleb is at his best in his philosophical asides and anecdotes involving Fat Tony, Socrates, and other characters.

One memorable concept in Taleb’s overall work is “domain dependence”, the failure to transfer competence from one domain to another – there is the story of the businessman paying the porter to lift his luggage, only to lift weights in the hotel gym ten minutes later. Taleb doesn’t suffer from domain dependence, in fact he has the opposite, which (in the spirit of his book) is not the ability to see connections between related fields, but the compulsion to draw analogies between any two things, sometimes even when there aren’t any or when they are obvious. Overall, I think the book contains many deep and counterintuitive observations, it gives a powerful perspective of looking at phenomena in terms of convexity or concavity of payoff, as well as excellent and usable practical philosophy. As all of Taleb’s writing, it challenges assumptions and provokes thinking – the fact that some parts are thought through less than others reduces its value only marginally.

Lucian Boia: History and Myth in Romanian Consciousness. This review refers to the Romanian original, “Mit şi istorie în conştiinţa românească”, which has been a bestseller in Romania and is credited with starting a current of “demythologization” of historiography. It has provoked heated debates and prompted adjectives ranging from the “courageous” and “first-of-its-kind” to the “unscientific” and “unpatriotic”. Considering its reputation, the book does relatively little myth-busting. The author finds questions such as “what really happened eight hundred years ago” less interesting than those of the type: “what did people a hundred years ago think about what really happened eight hundred years ago”. Questions of the second kind are easier to answer, and as Boia argues, they often hold more relevance to who we are and how we think today than questions of the first type.

As such, the book is not so much a Romanian history, but rather a “metahistory”, a history of Romanian historiography, which might sound like a dry topic, but the book is anything but – Boia describes in an entertaining, often subtly ironical style the mechanisms of myth-making, and traces how the dominant views of historical events evolved throughout the ages. These views were influenced partly by the intellectual currents of the time, partly by the political realities and necessities of each era. Especially interesting is the study of how the perceived significance of historical figures fluctuated throughout the ages.

The mythologization of history is not unique to Romania of course, but Romanian history does seem to offer a combination of a scarcity of resources for certain periods (resulting in a relative lack of consensus among historians), and a tendency (especially during the later part of the communist era) of considering historiography a tool for “nation-building”, legitimization of the regime, and a means of diverting attention from everyday (economic, social) problems. As Boia vigorously points out (and as some of the reactions to the book illustrate), the legacy of these trends largely lives on today. Boia’s book thus makes the important step of presenting historical myths for what they are, and treating them unemotionally, as prime objects of study. Due to the fine observations, intimate knowledge and sympathy towards the sources, and intellectual honesty displayed, the book is an excellent read, although possibly slightly confusing for someone not immersed in Romanian history.

Tony Hsieh: Delivering Happiness. An autobiography of sorts of successful web entrepreneur, Tony Hsieh. The first part of the book is more personal, telling the story of the founding of his companies, different business decisions, ups and downs of running the businesses and some personal life events. The tone is lighthearted and the stories are funny. The second part focuses more on the approaches in managing, and includes a codified “company culture” document with ample discussion on its margin. While the ideas about fanatical customer support and encouraging group cohesion within the company are interesting and probably of practical value to entrepreneurs, as the book progresses, the tone becomes disturbingly cult-like – despite claiming the opposite, the text steers into a pretentious and generic business mumbo-jumbo.

Avinash K. Dixit, Barry J. Nalebuff: Thinking Strategically. The book sets itself the goal of analysing elements of strategy in the abstract with illustrative examples from history, politics, business, sports, movies, or parenting. This might sound like a lofty approach, but the book is in fact excellent, and the examples are analysed with insight and humour. The book can be read as an introduction into elementary game theory, and it is one of the most engaging attempts at this that I have seen. It explains with clarity concepts such as sequential and simultaneous games, mixed strategies, the basics of voting systems, bargaining theory, and incentive schemes. While the book is readable for a wide audience, it doesn’t shy away from including some mathematical details when describing the structure of certain systems.

Alternatively, the book can be read as a collection of practically useful tactics and tricks one can employ in diverse situations. I find it hard to imagine how one could memorize all such “patterns”, but some simple concepts, such as that of “brinkmanship” are certainly memorable.

Part 7

Johan Huizinga: The Waning of the Middle Ages. (originally written in 1919)
This beautiful and influential book paints a broad picture of the Late Middle Ages, describing a world of contrasts and extreme passions, in which a feeling of decadence and insecurity, a “sombre melancholy” permeates everything. In Huizinga’s Middle Ages religious excess, solemn rituals, ceremony and pompousness are omnipresent but slowly becoming superficial, turning into parodies of themselves. In Huizinga’s words, the culture of the era is “overripe”, about to be fundamentally transformed (although Huizinga sees the transition to Renaissance differently from many other authors). Behind the carefully maintained idealized facade, we get occasional glimpses into the violent realities of life in the Middle Ages – often irrational, sometimes grotesque or comical.

The main premise of the book is that contrary to popular imagination, people in the Middle Ages had a worldview and a set of ideals and aspirations vastly different from ours. The book explores many aspects of this worldview, pertaining to love, religion, politics, art, life and death. After convincingly arguing and illustrating this thesis, in the end, the book still leaves us with the feeling that at a deeper level, or at least in its fragments, the mentality of the Middle Ages was not so alien afterall. The book is insightful and opinionated, the descriptions are vivid, they rely on ample documentation, and the arguments are compelling. The author is at his best when interpreting works of art. However, given the goal of capturing something as fuzzy as the general mood of an era six hundred years ago, the book is perhaps closer to an extended literary essay than to a precise scholarly work. Many of the passages are thought-provoking, although many of the ideas have long become part of the mainstream narrative about the Middle Ages in the time since the book was written.

Andrew S. Grove: Only the paranoid survive. The author shares his insights about the IT industry and about working in technology, drawn from his successful career as CEO of Intel. Interesting topics include the story of the pentium bug, the decision of Intel to focus on microprocessors instead of memories, the transformation of the computer industry from vertical to horizontal stratification (although, arguably, later the opposite has also been happening). The recurring theme is that of “inflection points” in technology, periods of fundamental change, in which, well, “only the paranoid survive”. The author describes subtle signals that hint at an impending change and describes strategies for dealing with it. Drawbacks of the book include the slight repetitiveness, and somewhat self-congratulatory tone, but overall I found the book insightful and also prescient for a work written almost twenty years ago.

Elizabeth Marshall Thomas: The Harmless People. We get a glimpse into the life of the Bushmen (also known as San people) in the Kalahari Desert during the 50s, when the author spent long periods of time living among them. The book describes in a simple, sympathetic manner the Bushmen’s everyday life, their hunting-gathering activities, their family life and customs, their music and celebrations, as well as their views of life and the world. There’d be little to be unsympathetic, observing the simplicity, gentleness and naivety, as well as the extensive practical knowledge about the surrounding natural world of these people. The book is beautifully written, the author avoids any trace of condescension or stereotypical portrayal – she observes the characters respectfully and lovingly and describes them in a precise and vivid way. In the last chapter the author revisits the places and people thirty years later, to find the world she earlier described in an irrevocably transformed state. She finds this transformation to be mostly for the worse: the lives of most of the characters have taken tragic turns, the traditional way of life has proven powerless against the encroaching modern society, and the implicit social order previously observed has broken down. Despite the sobering analysis, the book ends on a note of optimism about the future.

Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos, Annie Di Donna: Logicomix: An Epic Search for Truth. A writer, a computer scientist, and two illustrators make a comic about the foundations of mathemathics, telling embellished stories about the lives of Bertrand Russel, Georg Cantor, Kurt Gödel and other great logicians, with cameo appearances by the authors themselves. How can the end result be? The answer is: surprisingly (or predictably) good.


Did you read some of these books and have a different opinion? Let me know in the comments.

What was the best book you read in 2013?

A puzzle of apples and oranges

I first read the following wonderful little puzzle on Mathoverflow, where it was posted by David Futer. I don’t know who invented it or where it appeared first – if you have any information, please let me know. (Note: Toshihiro Shimizu points out the following link: AoPS)

There are 99 bags, each containing some number of oranges and some number of apples (any of these numbers can be zero). Show that we can always select 50 bags out of the 99, such that these 50 bags together will contain at least half of all the apples and at least half of all the oranges.

I like this puzzle because there are many different ways to solve it. Before reading on, do yourself a favor and try solving it yourself – it is not difficult, but very rewarding. If you find a proof that is different from those below, let me know and I’ll add it to the list.


Why look for different solutions, once we already have one? Partly because it’s fun, partly because each solution might give some new insight into the problem, partly because we can showcase different mathematical techniques, and perhaps most importantly, because each new proof lends itself to generalization slightly differently – so a new proof might lead to new results. The proofs below are sketchy in places, in order to keep them short – it can be instructive to try to fill in the gaps, or to try coming up with the proof after reading just the name of the technique.

Sometimes proving a more general statement is easier (at least psychologically – by removing distracting elements). In our case, it will be more “fruitful” to prove the following statement:

Given \(n\) bags that contain oranges and apples, we can select \( \lceil \frac{n+1}{2} \rceil \) bags such that they contain at least half of each type of fruit. (we only really care about the case when \(n\) is odd)

Here are all the proofs that I am aware of.

Proof 1. (sorting) (source: I vaguely recall seeing this proof on a math competition website but I can’t find it anymore. If you have any information, please let me know see AoPS link above).

Sort the \(n\) bags in decreasing order of the number of apples (break ties arbitrarily). Select the first bag, and from each following consecutive pair select the one with more oranges (break ties arbitrarily).
It is easy to see that the selected bags satisfy the required properties.

Proof 2. (induction) (source: idea from David Futer).

First we show that for \( n=3 \) the statement is true. Indeed, we can select the bag with the most apples and the one with the most oranges (if the two are the same, we also pick another bag).
Assume that the statement is true for \(n-2\). We then prove it for \(n\):
Consider a “good split” of the \(n-2\) bags into A and B, such that A has at least half of both fruits in A+B (such a split exists by the induction assumption). Now we try adding the other two bags, x and y.
If A has as many of both fruits as B+x or as B+y, we are done (we add the other bag to A).
Otherwise if B+x or B+y have as many of both fruits as A, we are done (we add both bags to B).
Otherwise if A has less apples than B+x and less apples than B+y we are done (add to A the one of x and y with more apples). Symmetric argument for oranges.
Otherwise if A has less apples than B+x and less oranges than B+y, it means that A has less of both fruits than B+x+y, which we treated already. Symmetric argument after swapping the terms “apples” and “oranges”. There are no other cases.

Proof 3. (contradiction or “canonical counter-example”)

Suppose there is no solution, then consider a set of \( \frac{n+1}{2} \) bags which has at least half of the apples (there is always such a set), and among such sets the one with the most oranges possible.
Let this set be A, and the remaining set B (A has at least half of apples and B has more than half of oranges).
Take the bag from A with the least oranges and move it to B. Now take the bag from B with the most oranges and move it to A. If we just moved the same bag back, it means that all bags originally in B have at most as many oranges as the bag with the least oranges in A. This contradicts the fact that B has more than half of all oranges.
Otherwise there are two cases: 1) the number of apples in A drops below half when we move a bag, which means that B becomes a solution, or 2) the number of apples in A remains at least half throughout, then after the swap we increased the number of oranges in A, which contradicts the statement that the chosen set had the highest such value.

Proof 4. (counting) (source: I read this proof written in probabilistic form by Erez Druk, after being asked by Dror Ozeri on a puzzle forum – as it is a closed forum, I cannot link to it)

I write this and the next proof in the n=99 case only, to avoid excessive notation. The number of 50-sets of bags is the same as the number of 49-sets \({99 \choose 49}\). Denote the number of 50-sets that contain at least half of all oranges by \(N_{50}\), and the number of 49-sets that contain at least half of all oranges by \(N_{49}\). We have \(N_{50}>N_{49}\), unless none of the bags contain any oranges, which is an easy case anyway (there is one-to-one mapping from 50-sets to 49-sets, such that each 50-set contains its mapped 49-set). Clearly, more than half of all 50-sets contain at least half of all oranges (otherwise the previous inequality would be violated: if an 50-set does not contain at least half of all oranges, then its complementing 49-set does).
The same argument holds for apples. As more than half of the 50-sets contain at least half of the oranges, and more than half of the 50-sets contain at least half of the apples, there is at least one 50-set containing at least half of both fruits.

Proof 5. (price function) (source: from David Futer)

Assign price \(x\) to an orange, and \(1-x\) to an apple. Select the 50 most expensive bags.
\(f(x) = (\mbox{# of oranges in top 50}) / (\mbox{total # of oranges})\)
\(g(x) = (\mbox{# of apples in top 50}) / (\mbox{total # of apples})\)

We have \(g(0) \geq 1/2, f(1) \geq 1/2\), and \(f\) is increasing, \(g\) is decreasing. Clearly there is some \(x\) for which both \(f\) and \(g\) are at least 1/2, which denotes a solution (if for some \(x\) both functions would have values below 1/2, then the top 50 would not be the top 50, as its complement would have more apples and more oranges, therefore higher total price).

Proof 6. (geometry – Ham-Sandwich) (source: from David Futer, and a variant explained by Dror Ozeri on the same forum)

We represent the bags as sets in the plane, with the restriction that a straight line can cross at most two of them (this can always be achieved). We somehow encode the number of oranges and apples in each bag. There are more possible variants here: map each fruit to a red or blue point (depending on its type); or map all oranges of a bag to a single point with a weight that tells the number of oranges, and similarly for all apples of a bag; or map the whole bag to a single point for which we define an “apple-measure”, and an “orange-measure”. Now we apply the Ham sandwich theorem to obtain a balanced cut (there are different variants of this theorem, we choose the one suitable for our representation). If any “bags” are cut by the obtained line, we move them to the side which has less bags. This gives a valid solution.


  • Prove the following more general statement. Which proof technique is suitable?
    Given \( n\) bags that contain \( f\) kinds of fruits, we can select \( \lceil \frac{n+f-1}{2} \rceil \) bags that contain at least half of all kinds of fruits.
  • Prove the following variant. Which proof technique is suitable?
  • Given 100 bags that contain apples and oranges, we can select 34 bags that contain at least one third of the apples and one third of the oranges.

  • Find a general statement with \(n\) bags, \(f\) kinds of fruits and finding a subset of the bags that contains at least a \(p\)-fraction of the fruits of each kind, for arbitrary \( 0 \le p \le 1\).
  • Assume that we know the contents of each bag. Which proof leads to an efficient algorithm for finding a good subset of bags in the sense described above? In particular, which proof leads to a linear time \(( O(n) )\) algorithm?
  • Can you come up with new proofs for the puzzle? If so, I would love to hear them. Do any of the proofs admit new generalizations?

Thanks to David Futer, Lavinia Dinu, Manuel Arora, and Tobias Mömke for interesting discussions about this puzzle.

Nonlinear speedometer

As everyone knows, excessive speed is one of the main causes of traffic accidents. One of the reasons for reckless speeding is that when we think of the possible impact of a collision, our intuition fools us. We tend to assume that if we go twice as fast (say, at 80 km/h instead of 40 km/h), then the impact of a collision will be twice as large. In fact, the impact of a collision is proportional not to the speed (V), but to the kinetic energy of the vehicle (m/2 * V²), which is transformed to work as the car decelerates to zero. Here m is the mass of the vehicle, which we cannot do much about, but the other term is the square of the velocity. This means that the “impact” of hitting a wall at 80 km/h is four times as large as that of the same collision at 40 km/h.

To capture this intuition, the idea of this post is a speedometer design, that scales as the square of the velocity, to give the driver a more realistic view of the effects of speeding.

nonlinear speedometer
(larger image)


  • after sketching this drawing, I found that in 1995 Goetz Kluge was thinking along similar lines and produced a similar design. There are however some differences – see next.
  • besides showing the speed in the more familiar, circular layout, I also made the decision to draw the smaller lines at uniform density (i.e. they are spaced proportionally to the kinetic energy, not to the velocity). This design might make it harder to estimate the exact speed, say between 40 km/h and 50 km/h, but it makes the increased kinetic energy easier to grasp – when we speed from 120 km/h to 130 km/h, the hand crosses more “small lines” than from 20 km/h to 30 km/h.
  • while I haven’t seen a similar speedometer in any real car, some of the existing speedometers are in fact nonlinear. Unfortunately, they seem to achieve the exact opposite effect to the design shown above. See this example. On the linked image, and in many other modern speedometers, the manufacturers try to put more resolution in at the lower half of the range, dilating the velocities between 0 and 80. My guess is that car makers do this, because the car accelerates faster at low speeds, so dilating the lower range makes the hand movement seem more uniform at all speeds – this is probably aesthetically more pleasing, but it might come at a cost of an increased (false) sense of security, and in effect a reduced safety.

    (I got this exactly wrong – thanks Tobias for pointing it out – it is actually the opposite, on the proposed speedometer design, the hand would move more evenly – so besides being safer, it would be more aesthetical as well, the only downside seems to be the reduced resolution in the low range)

Map of named colors

In this post I present a visualization of all named colors, ranging from the more common to the more obscure, including green, blue, yellow, orchid, lavender, turquoise, acquamarine, azure, etc. etc.

From various sources [1] I downloaded around 2600 such names, I merged the different lists, and slightly edited the entries to remove duplicates and to make the collection more consistent. Then I trained a SOM (Kohonen map) [2] with 6000 units, using the open source toolbox [3] for Matlab. Finally I exported the results with some custom scripts I wrote. This last part was done similarly to the visualization of languages described in my previous post. Otherwise the data was much more reasonable this time than in the previous post: 2600 data points, 3 features (red, green, and blue values) and no missing entries.

The first plot shows the trained SOM map with each hexagonal unit being colored according to its location in the three-dimensional RGB space. The other three maps show only the red, green, and blue values of the units, respectively.

exp_som.png (~6 MB)
Figure 1. SOM map of named colors.

The second plot contains the actual names of the colors and the corresponding hexadecimal RGB values. The goal was to place similar colors near each other as much as possible.

exp_som_text_scale.png (~12 MB)
exp_som_text.svg (~3 MB) (NEW: svg format)
Figure 2. Map with color names.


Name That Color project (and references therein) by
Chirag Mehta.
Wikipedia: List of colors.

Color Space Dimensionality Reduction project by Aubrey Jaffer and references therein. This project includes two dimensional maps of colors created using different techniques.

In this project SOM is used to visualize colors, but only a handful of colors are shown.

Some of the colors in the final list I used seem to have been submitted by users at the Colourlovers website.

[2] Scholarpedia:

[3] SOM Toolbox:

Visualizing the languages of the world

I recently came across the WALS (World Atlas of Language Structures) data set [1] which contains structural information about 2676 languages and dialects [2] from all around the world. The WALS website contains a detailed explanation of all the features and it also shows on the world map the geographical distribution of the languages and of the different features. As soon as I saw the data I started thinking about how to visualize it differently in order to highlight similarities between languages and language families.

When laypeople (myself included) naively think about similarities between languages, they mostly consider similarities between words (such as between the English “brother” and German “Bruder”). As the WALS data set ignores the vocabulary, and describes instead the deeper, structural aspects of language, I was hoping to obtain some interesting visualizations.

This post describes in detail the steps I took to visualize the data. If you want to see the resulting images only, scroll down to the end.

1. The data set

The WALS data set contains 2676 languages and 192 features, mostly related to phonology and grammar. For example, one feature describes whether a language has one, two or three grammatical genders, another feature refers to the inflection of nouns, yet another refers to the usual position of the verb in the sentence, and so on.

The features seem to have been meticulously collected and curated, however, many values are missing. More precisely, out of roughly half a million possible entries, only 15% are available. It is not clear whether a missing feature means that it has not been observed, or that it does not make sense for a given language. I assume it is mostly the first case: for languages spoken by very few people it must be difficult to collect reliable data and on some languages there are few publications available. However, there are also cases when a feature is not applicable to all languages. For example, one of the features is “Nasal Vowels in West Africa” which is missing for all languages outside of West Africa. It would improve the analysis if we could distinguish between these two types of missing values, but as I had insufficient information to do this labeling myself, I simply considered all missing values of the same type.

Besides the structural description of the languages, the data set contains the following meta-data: the classification of languages into families, subfamilies and genera, the ISO-code of each language and the geographical location where the language is spoken. The latter information is not always useful, as it points to a single location. In case of Russian, for example, this location is Moscow. For Portuguese, there is no separate entry for Brazilian Portuguese, yet the geographic location indicates only Lisbon.

2. Preprocessing

Looking into the data, one thing that can be observed is that a few languages share the same ISO-code – these seem to be closely related dialects. When this is the case, it seems that only one of the languages has the full description, and for the others only the differences are encoded. Therefore, as a first preprocessing step I copied over features between pairs of languages that have the same ISO-code, whenever a feature was present for one, but not for the other language. Overall, this operation affected the proportion of missing values only marginally (about 100 languages of the 2676 have had some feature copied over). However, this again raises the question of the exact meaning of missing data. If there are other cases when some features are missing because they are “too obvious”, that could distort our analysis – however, we have to live with this uncertainty for now.

As a next preprocessing step, I “binarized” the data. All the features in the data set are “categorical”. For example, feature #52, called “Comitatives and Instrumentals” [3] has the following possible values: 1 for “Identity”, 2 for “Differentiation”, and 3 for “Mixed”.
Without getting into the details of what this feature (or other features) really mean, it is clear that the assignment of numerical values is quite arbitrary here. Therefore, to perform numerical operations on the data further on, it made sense to split this feature into three. The new features (called 52.1, 52.2, 52.3) encode whether the value of this feature is 1, 2, or 3. (for example, a value of 2 is transformed into 010 and 3 is transformed into 001).

Some of the features, however are of “ordinal” type. For example, the first feature has the following five values: 1 “Small”, 2 “Moderately small”, 3 “Average”, 4 “Moderately large”, 5 “Large”. In this case (and for some other features of such type) it is more sensible to binarize the data as follows: 1 becomes 10000, 2 becomes 11000, …, 5 becomes 11111. That is, we split the feature into different parts that encode whether the value is larger than some threshold. I binarized all features in one way or the other (overall I used the “ordinal” method for 12 features of the 192).

I performed these preprocessing steps with some short Matlab scripts I wrote, and in the end I obtained a binary data set with 1141 columns and 2676 rows and (still) most of the entries missing.

3. Dimensionality reduction

Besides the missing entries, the biggest obstacle in the way of visualizing the data is that it lives in 1141 dimensions. As a first step, I reduced the number of dimensions to 30, to make the data easier to handle. Typically, this is performed using a linear projection method called principal component analysis (PCA) [4]. When some data is missing, usually a simple “imputation” method is used, such as deleting the columns or rows with missing entries, or replacing the missing data with column averages. However, when such a large portion of the data is missing, these simple tricks are insufficient.

Instead, we can use a nonlinear version of PCA, tuned especially for the case with many missing values. This method, while reducing the number of dimensions, also attempts to “fill in” the missing values. In this case, the conceptual simplicity and efficiency of PCA is mostly lost, but the methods are still quite efficient. The theory and assumptions behind such methods are described in this paper written by A.Ilin and T.Raiko [5] (disclaimer: former colleagues). Implementations are available in an open source toolbox for Matlab [6] developed by the same authors.

Using PCA when the data is binary is not quite the optimal choice, and there exist variants especially tuned for binary data (see for example our 2009 paper [7] for some theory and experiments), however, as my goal here was not prediction or learning, just visualization, I decided to stick to the simpler methods.

After this operation, we are left with only 30 “aggregate” features and no missing data. We are now ready to visualize the data.

4. Visualization

I first decided to look at some very small subset of the data, therefore I selected the 24 rows corresponding to Romance languages [8] according to their “genus” label. To visualize these data points, I trained a SOM (Kohonen map) [9], using the open source toolbox [10] available for Matlab. This method embeds a two-dimensional grid into the higher dimensional (in this case 30 dimensions) space, such that the grid is “close” to the data points. At the same time the neighbors in the grid exert some “pulling” on each other, which serves as a kind of regularization. In the end we can visualize the grid points in the so-called U-Matrix. For visualization I wrote some custom scripts that plot the output that I saved from the SOM toolbox.
Figure 1. Romance languages U-Matrix.

Here the grid is hexagonal and has 40 cells. On the image we can see each of the 24 data points (Romance languages) mapped to the closest grid cell (in the 30 dimensional space). The color of a cell is indicative of the distances from neighboring cells (brighter color means larger distance), in this way if there are clear clusters, they will appear as dark valleys separated by bright “ridges”.

Our hope with the visualization is that pairs of languages that are similar would be mapped nearby and pairs that are different would be mapped far away. Unfortunately, in general, not all of these constraints can be satisfied, so we can’t always draw conclusions from the fact that two languages appear close to each other. This is where the coloring of the U-Matrix should help. In many cases, though, it seems that proximity does indeed correspond to linguistic similarity (here, the dialects of Romansch [13] are mapped together, just as Romanian together with Moldavian, etc.)

To get some intuition on how the languages were laid out on the map, I made a plot of the original features on the same map. This is on the next image. There are 192 copies of the same hexagonal map (one for each feature). The locations of the languages are the same as in the U-Matrix. On these maps the value of the feature is indicated for each language. The descriptions of the features can be found on the WALS website [11]. Looking at these maps we can check for two languages that appear nearby, on which features they agree. Black is for “missing data”.
Figure 2. Romance languages feature map.

Let’s also plot the geographic location of the languages on the same map. Note however, that this information was not used during the training phase. Here again the languages appear in the same positions as before, and their geographic location in degrees is indicated by color. The goal of this plot is to see if there is some correlation between geographic distance and linguistic difference. We can read off some obvious facts from the plots, such that Canary Islands Spanish is both the westernmost and the southernmost, and Moldavian is the second easternmost after Ladino [13]. Otherwise the number of data points is a bit too small here to observe interesting correlations.
Figure 3. Romance languages Lat/Lon.

On the two previous types of visualizations a hexagonal cell is colored with a single color if there is a single language mapped to it, or if all languages in that cell share the same value for a feature. Otherwise a small pie-chart is displayed showing the share of each value.

Now let’s look at a larger subset. I selected all languages from the Indo-European family of languages [12], according to their “family” label. This meant 176 rows in the data set. The visualizations are similar to the earlier ones, but now I also made an extra plot showing the different genera within this family. Here the black color shows whether a language belongs to a given genus. It can be observed that the clustering on the SOM is in many places consistent with the classification information. This means that the usual taxonomy of languages is largely consistent with the structural linguistic data, which is hardly surprising. There are however, some unlikely pairs of languages placed close to each other on the map. One could examine the data to check in each case whether there is really some structural similarity, or we can simply write it off as noise. We can also observe a quite sharp East/West separation between languages, and a bit less pronounced North/South separation (the latter picture is complicated by an outlier – the southernmost language of the family is Afrikaans [13]). For many of the features a nice clustering of the values can be observed.
Figure 4. Indo-European languages U-Matrix. (large file)
Figure 5. Indo-European languages feature map. (very large file)
Figure 6. Indo-European languages Lat/Lon.
Figure 7. Indo-European languages genera.

Finally, let’s visualize all languages from the data set in a similar way. To restrict attention to spoken languages only, I removed the 40 sign languages from the data set. There remain 2636 languages. The resulting maps are very large, but quite informative, again, there is an apparent clustering that closely resembles the known taxonomy of languages, but other similarity relations can also be observed on the map. As the maps became very large, here I only include the U-Matrix.
Figure 8. All languages U-Matrix. (very large file)

Please note that two languages can be placed nearby even if there is very little similarity between them, especially if one of the languages has very few features filled in. This is a constraint of the method used, but also inherent to the problem itself – we lose information by reducing the number of dimensions. So please think twice before using these plots for drawing conclusions about the relatedness of very distant exotic languages or for supporting such theories :)

To (partially) overcome the previously mentioned problem, I also plot the data when restricted to languages having relatively few missing entries. This also makes the maps somewhat more managable in size. Here are the plots for the data sets having less than 160 missing features.
Figure 9. All languages with sufficient data: U-Matrix.
Figure 10. All languages with sufficient data: Lat/Lon.
Figure 11. All languages with sufficient data: families.

All figures: [link to album]


[1] Dryer, Matthew S. & Haspelmath, Martin (eds.). 2011. The World Atlas of Language Structures Online. Munich: Max Planck Digital Library.

[2] “A language is a dialect with an army and a navy” – Max Weinreich
The question of what constitutes a language and the distinction between language and dialect is a difficult, and often emotionally and politically charged question – for example, in this data set Romanian and Moldavian appear as different languages, whereas many consider them to be too similar even to be considered separate dialects.

[3] Stolz, Thomas & Stroh, Cornelia & Urdze, Aina. 2011. Comitatives and Instrumentals.
In: The World Atlas of Language Structures Online. Max Planck Digital Library, chapter 52.


[5] Ilin, Raiko: Practical Approaches to Principal Component Analysis in the Presence of Missing Values, JMLR, 2010.

[6] PCA with Missing Values software:

[7] Kozma, Ilin, Raiko: Binary Principal Component Analysis in the Netflix Collaborative Filtering Task, MLSP, 2009.


[9] Scholarpedia:

[10] SOM Toolbox:

[11] WALS features:


[13] Romansch:

Disk intersection game

Here’s a small puzzle/game that I made, using a simple geometric concept. You can try it here: disk intersection game. The game idea is loosely inspired by the game planarity, but it is also significantly different.

disk game screenshot

There are some disks, and you can move them around arbitrarily (drag with the mouse). The goal is to make all of them “good”. When a disk is “good”, its color turns to green. The other colors indicate different levels of “badness”, as shown in the legend to the right. With some experimentation you can probably discover what you need to do to achieve this, but here is an explanation of the idea:

There is a hidden “model”, created randomly, which you need to discover. The model determines how the disks need to be interconnected, meaning which pairs of disks have to intersect each other (overlap with each other). If you “realize” this model, by moving the disks in such a position as to intersect each other in the required way, you win the game. The colors of the disks indicate how far the current configuration is from the correct model. This is computed as follows: if a certain disk should intersect disk A and no other disks, but instead it intersects disk B and C, and it doesn’t intersect A, then its “badness” is 3, so it has the color corresponding to the number 3. In other words, “badness” shows how many errors there are due to a given disk. When the “badness” of all disks is 0, you have solved the puzzle.

If you find the color codes hard to recognize, you can click the “#” button, and the “badness” will appear as a number. You can also ask for help three times during the game. Clicking the help button will display the model that has to be realized (i.e. the connections between the disks, shown as lines). There are easy and hard games, in a hard game the model tends to be more complicated, therefore harder to discover.

Try the disk intersection game.


  • I came up with the idea for the game, while working on a problem related to unit disk graphs. This is the first iteration of the game, so probably the gameplay and the design could be significantly improved. If you have any suggestions, please let me know in the comments.
  • There is nothing special about disks here, maybe other intersection graphs would make more interesting (or more challenging) games.
  • One reason I wrote the game, besides wanting to try out the idea, was to get familiar with HTML5 Canvas – in the future I’d like to make another, somewhat more complex game. As this was my first encounter with Canvas, and because I wrote the game in one sitting (~3 hours), the implementation is a bit rough around the edges. Please let me know if you find any bugs or if you have suggestions.

Books I read in 2012

Here is the list of books that I read (in English) during 2012, or at least those that I found interesting enough to describe in one or two sentences each.

Part 1

Philip K. Dick: Do Androids Dream of Electric Sheep? The inspiration behind the Blade Runner movie, including replicants and the Voight-Kampff test but excluding the Tannhauser Gate and C-Beams. Wonderfully coherent science fiction, both deeper and psychologically better motivated than the movie.

Andre Agassi: Open: An Autobiography. Entertaining and motivating. The inner dialogues during tennis matches are especially interesting, as is the (one-sided) description of the rivalry with Sampras. His great comeback is inspiring and fun to read, the tone becomes slightly preachy towards the end.

Jessica Livingston: Founders at Work: Stories of Startups’ Early Days. Interviews with famous and obscure founders of companies. The selection is excellent, the interviewees are diverse, and each interview touches on some different aspect of startup-life, although there are many recurring themes. The book is consistently good, radiating with optimism.

Jack Kerouac: On the Road. A book that needs no introduction, obviously. Probably it means different things to different people, I just found it thoroughly entertaining and the style refreshingly sharp and concentrated. The characters in the plot seem to be searching for the ultimate experience with varying levels of success.

Part 2

Wilhelm Reich: The Mass Psychology of Fascism. Unlike Reich’s later, controversial stuff, this book is lucid and coherent, the main theses are quite convincingly argued. The strictly historical account is interesting in itself, but the book also explores more general themes, like the connection between the appeal of totalitarianism, fake morality (especially around sexuality), irrational mysticism. (1933)

John Maynard Keynes: The General Theory of Employment, Interest and Money. As Keynes appears to be as influential as ever, this book is probably more often debated than actually read. Which is a pity, although not an easy read, the book is written with amazing clarity. The style is elegant and old-fashioned, academic and precise: Keynes is careful to delimit the range of applicability of what he says (I suppose most detractors take his ideas outside of this range). But once you accept the boundaries he sets, there is a certain inevitability to his claims: given this and that, “ceteris paribus”, this and that relation has to hold between this and that economic quantity. The book is worth reading for the precise definitions of economic terms alone. It would be foolish to claim that I fully understood (let alone retained) a large fraction of the ideas in the book, and I did skip pages in some of the chapters that I found less interesting, but overall, it was a joy following through the arguments. (1936)

F. A. Hayek: The Road to Serfdom. Just as in the case of Keynes, the debates surrounding this book seem to be much less interesting than the ideas in the book itself. First, one would assume that the advocation of “economic liberalism” would be at odds with “keynesianism”. However, Keynes himself warmly praises the book in his review. Here, the main theme is the relation between individual and state. Hayek convincingly argues that the difference between the political “far left” and “far right” is less interesting than the difference between totalitarianism and classical liberalism. Hayek derides many forms of centralized planning, and describes the harmful mechanisms set in motion by them, as well as the transition from an all-is-allowed-except-… to an all-is-forbidden-except-… society. The historical accounts, particularly about Germany, are insightful, however, his predictions seem to have been either mistaken, or at least on a different timescale than plausibly assumed (or perhaps too influenced by the times in which the book was written). (1944)

Primo Levi: Survival in Auschwitz. Originally titled “If This Is a Man”, the book relates the arrest, incarceration and eventual liberation of the author from the Auschwitz death camp. It describes the dehumanizing experience in an understated, precise way. Despite the tragedy beyond description, the book offers a vision of hope and humanity.

Part 3

Rosamund Stone Zander, Benjamin Zander: The Art of Possibility. A very nice collection of anecdotes and distilled advice, mostly about how to defuse conflicts and how to look at difficult situations from a new perspective. The authors draw from their vast experience in music, education and counselling/consulting. It does get too self-congratulatory at places, but as far as self-help-feel-good-type books go, it is surprisingly good.

Victor Pelevin: Omon Ra. Quirky, cynical, intelligent science fiction. At the surface, a satire of the Soviet space program (or rather, of its caricature), in reality, a more universal tale about aspirations, ideology, and the human condition.

George Orwell: Animal Farm. Again, a classic about which it is impossible to say anything new. Despite being intented (apparently) as a satire of events in a concrete place and time, the book is as relevant and as universally accessible today as ever. (And this isn’t something new to say about it either.)

John E. Littlewood: Littlewood’s Miscellany. “The surprising thing about this paper is that a man who could write it – would.” The book contains many similar quips, more or less connected to mathematics or the process of doing mathematics. There are also many interesting and deep mathematical puzzles (some of them have become classics since the book was written), linguistic paradoxes, short anecdotes about mathematicians, amusing mathematical errors, unintentionally funny formulations from books or papers, references to contemporary results. The book is a collection of fragments assembled by Littlewood, refreshingly lacking any serious organization. Written in 1953, it seems to have stood the test of time, the few parts that feel outdated do so only because of the attention to detail of the author that seems uncommon today.

Part 4

I.M. Gelfand, Mark Saul: Trigonometry. It is often claimed that in order to effectively teach a topic, one needs to know more than what is being taught. Taking this advice to the extreme, here one of the great mathematicians of the twentieth century teaches early high-school level trig. The result is predictably pleasing: the right mix of formal clarity and informal insight, it seems to anticipate and answer any question the reader might have and provides the right level of challenge at every point. If this much thought went into every textbook, the state of math education would be quite different. Part of a series on high school math by the same author.

G-C. Rota: Indiscrete Thoughts. A wonderful collection of essays loosely related to mathematics. The first part contains anecdotes about famous mathematicians of the twentieth century, often focusing on their character flaws and amusing aspects of their life – part of the reason for “indiscrete” in the title. Most interesting – and perhaps least indiscreet – is a moving tribute to Stanislaw Ulam, mixed with nostalgic observations about pre-war Central Europe. The second part contains various essays about topics such as the philosophy of mathematics and science, mathematical discovery, philosophy of the mind, aesthetics in mathematics, etc. While less colourful and entertaining than the first half of the book, Rota manages to keep it readable and interesting for the most part. The book ends on a lighter tone, with observations about concrete mathematical fields, tips on how to do mathematics, and career advice of sorts ranging from time-management to creativity – refreshingly unconventional and thought-provoking.

G. Boltjansky, I. Gohberg: Results and Problems in Combinatorial Geometry. A beautiful small book containing three general geometric problems, which are shown to be intimately related. The discussion is entirely elementary, using only early high school level mathematics, but the book highlights deep connections between disparate fields and ends with difficult open problems.

Douglas Hofstadter: Metamagical Themas. A collection of Hofstadter’s columns for Scientific American. The essays are self-contained and easy to read. Due to the format, the author had less of a tendency of trying to tie up all loose ends, than in Gödel-Escher-Bach, maybe for this reason, I enjoyed the book much more than GEB. Some of the topics, such as the “prisoner’s dilemma” may have been new at the time Hofstadter wrote about them, but have become mainstream and have been discussed endlessly since then. Other topics include self-reference, typography, analogies, Rubik’s cube, sexism in language, consciousness, creativity, Lisp. Due to the format, the level of interestingness in the book is uneven, but I found most of the chapters stimulating and thought-provoking. The richness of ideas and the variety of topics make this my favorite book of 2012.

What was the best book you read in 2012?

Obstruction game

Some years ago I was reading about the game of Nim, then a bit more about the theory of impartial games, and as the rabbithole seemed to go deeper and deeper, I needed some pretext to continue. So I came up with the rules of a simple paper-pencil game that I could try to analyze with the theory that I was reading about. I never managed to fully analyze the game, but it seemed to be reasonably fun to play anyway. Later I found that some very similar games have already been described.

The rules are simple: two players take turns in marking squares on a grid (say, of size 8*8). You can only mark a square if all of its neighbors (including the diagonal neighbors) are empty. The first player unable to move loses.

More details of the game are here.

Recently, I had the pleasure of discovering a website that contains the descriptions of many paper and pencil games, most of them playable on the website against the computer. The site is maintained by David Johnson-Davies. In fact, I found out about the website from him, when he told me that he also included the game I described above. The game is filed under the name “Obstruction game” which very well describes what it is about. You can even play it against the computer. Here is the website with all the games and here is the link to the Obstruction game. There is even a description with some useful strategy-tips.

Happy playing!

Ordering cyclic graphs

Suppose there are n sports teams, some of them play each other, in each game one team wins and the other loses and in the end we want to order the teams such that if A beats B than A is before B in the order. If two teams A and B played each other multiple times but, say, A clearly dominated B, we consider as if they would have played once and A beat B. If they played each other several times and there is no clear winner, or if they played only one draw, we consider as if they would have played twice and once A won, once B.

We can represent the whole situation as a directed graph with n vertices {1, …, n} and an edge from i to j if i won against j. Clearly, there are no self-loops. If the graph has no cycles, we can efficiently compute a topological sorting of the vertices and we are done. What if the graph has cycles (two teams that played a draw, or a sequence of teams where A beat B, B beat C, …, X beat A), but we still want to output a reasonable ordering? What is a reasonable ordering?

This is the vertex ordering problem and we can formulate many different criteria. I’ll go through some variants and mention for each of them, whether they are NP-hard or polynomially solvable. It turns out that almost all of them are NP-hard, and those which are not are quite trivial (plus the topological sorting mentioned before, which is also polynomially – in fact, linearly – computable). I’ll try to refer to sources where I found some, where I didn’t find any, the result is simple enough to be considered folklore. Not all the criteria that I list are reasonable – a reasonable criterion should actually lead to the topological sorting when there are no cycles. Did I forget some sensible criteria? Let me know what you think.

The general set-up is the following: we search for a permutation f:[n] -> [n], such as to minimize or maximize some quantity as follows:

1. The first criterion is that we want to minimize/maximize some reasonable function of the number of wins and the number of losses of a team. This includes the methods used in most sport tournaments where wins and loses are worth some number of points and in the end the teams are ranked by their accumulated number of points. This criterion is easy, because we can just compute the indegree/outdegree of each vertex, compute the necessary function values and sort vertices according to it.

Now we move to some more interesting criteria.

  • We either minimize (MIN) or maximize (MAX) some quantity.
  • The quantity is either the SUM, the MAXimum, the MINimum or the COUNT of something.
  • That something is either the length of EDGEs, the length of BACK-EDGEs, or SIGNED length of EDGEs.

This gives 2*4*3=24 cases.

For example, if i->j means that there is an edge from i to j:

MIN COUNT EDGE refers to    

MAX SUM EDGE refers to    

MIN SUM BACK-EDGE refers to    

MAX MIN SIGNED-EDGE refers to     .

These are all for directed graphs, as mentioned earlier. Before starting to enumerate all these cases, let’s define the same problem for undirected graphs. For undirected graphs there are less cases, as we don’t have to consider separately any edge or back-edge or signed edge. Thus there are just 2*4=8 possibilities here.

Now let’s see what we’ve got, numbering continuously all the cases.

Undirected graphs

2. MIN SUM: This is called Minimum Linear Arrangement, a known NP-hard problem.

3. MIN MAX: This is called Minimum Bandwidth, a known NP-hard problem.

4. MIN MIN: makes no sense, it can always be 1.

5. MIN COUNT: makes no sense, it is always the number of edges.

6. MAX SUM: this is equivalent to 2. on the complement graph, therefore NP-hard.

7. MAX MAX: makes no sense, it can always be n-1.

8. MAX MIN: This is the anti-bandwidth problem, studied in literature, see here, or here (leads to paywall). It is NP-hard. This can be seen via the following reduction: “Does G have a Hamiltonian path?” is the same as “can the vertices of G be arranged such that there are edges between neighbors?” is the same as “can the vertices of G be arranged such that the complement of G has anti-bandwith at least 2?”. If we could answer the last question efficiently, we could answer the first.

9. MAX COUNT: makes no sense, it is always the number of edges.

Now we move on to the original question:

Directed graphs

10. MIN SUM EDGE: NP-hard. From 2. if we replace every undirected edge with a directed edge.

11. MIN SUM BACK-EDGE: NP-hard. From 2. if we replace every undirected edge with two directed edges.

12. MIN SUM SIGNED-EDGE: this is quite interesting. Notice that we can replace a path a->b->c with a single edge a->c without changing the sum. If we do this sufficiently many times we will be left with vertices with only incoming or only outgoing edges. To minimize the sum we clearly have to put vertices with outgoing edges first, then vertices with incoming edges. Otherwise we could make a simple replacement that would decrease the sum. Furthermore, vertices with outgoing edges have to be ordered in decreasing order of degree, vertices with incoming edges ordered in increasing order of degree. Again, otherwise, a simple swap would decrease the sum. That’s all.. Observe that what we’ve got is the same as ordering the original vertices in decreasing order of [outderee-indegree]. In this sense, this is a special case of 1.

13. MAX SUM EDGE: NP-hard. From 6. if we replace every edge with a directed edge.

14. MAX SUM BACK-EDGE: NP-hard. From 6. if we replace every edge with two directed edges.

15. MAX SUM SIGNED-EDGE: solution is the reverse order of 12.

16. MIN MAX EDGE: NP-hard. From 3. if we replace every undirected edge with a directed edge.

17. MIN MAX BACK-EDGE: NP-hard. From 3. if we replace every undirected edge with two directed edges.

18. MIN MAX SIGNED-EDGE: NP-hard. From 3. if we replace every undirected edge with two directed edges. Known as Minimum Directed Bandwidth.

19. MAX MAX EDGE: makes no sense, it can always be n-1.

20. MAX MAX BACK-EDGE: makes no sense, it can always be n-1.

21. MAX MAX SIGNED-EDGE: makes no sense, it can always be n-1.

22. MIN MIN EDGE: makes no sense, it can always be 1.

23. MIN MIN BACK-EDGE: makes no sense, it can always be 1 and has to be at least 1 if there are cycles.

24. MIN MIN SIGNED-EDGE: makes no sense, it can always be -(n-1).

25. MAX MIN EDGE: NP-hard. From 8. if we replace every undirected edge with a directed edge.

26. MAX MIN BACK-EDGE: NP-hard. From 8. if we replace every undirected edge with two directed edges.

27. MAX MIN SIGNED-EDGE: NP-hard. From 8. if we replace every undirected edge with two directed edges.

28. MIN COUNT EDGE: makes no sense, it is always the number of edges.

29. MIN COUNT BACK-EDGE: NP-hard, known as Minimum Feedback Arc Set.

30. MIN COUNT SIGNED-EDGE: makes no sense, it is always the number of edges.

31. MAX COUNT EDGE: makes no sense, it is always the number of edges.

32. MAX COUNT BACK-EDGE: NP-hard, it is the reverse order of 29.

33. MAX COUNT SIGNED-EDGE: makes no sense, it is always the number of edges.

Phew :) Did I forget any interesting criteria?

From the point of view of the original motivation (a ranking of teams) the criteria that seem to some extent reasonable are: 11, 12, 17, 18, 29.

Self-counting sentences II

Let’s revisit the topic of the previous post on self-counting sentences. We looked at sentences like this:

In this sentence the number of occurrences of 1 is __, of 2 is __, ..., of n is __.

Let’s go up one more step on the ladder of abstraction, to arrive at sentences of the type:

In this sentence, there are 3 numbers appearing 1 time, 1 number appearing 2 times, 1 number appearing 3 times, 0 numbers appearing 4 times.

In this sentence, there are 2 numbers appearing 1 time, 3 numbers appearing 2 times, 0 numbers appearing 3 times, 0 numbers appearing 4 time.

Observe, that both say the truth about themselves. Below are three puzzles. In all cases the task is to fill in the gaps such as to make the sentence say the truth and the questions are: for what values of n does a solution exist and what are the solutions (if they exist).

In this sentence there are __ numbers occurring 1 time, __ numbers occurring 2 times, ..., __ numbers occurring n times.

In this sentence, from the numbers 0,1, ..., n-1, there are __ numbers occurring underlined 0 time, __ numbers occurring underlined 1 time, ..., __ numbers occurring underlined n-1 times.

In this sentence there are __ numbers occurring underlined 1 time, __ numbers occurring underlined 2 times, ..., __ numbers occurring underlined n times.

Note: I submitted the first variant of the problem to Kömal, where it appeared in the 2013 September issue (Hungarian, English – unfortunately, as of this writing, the English version appears mistaken there). Later, Toshihiro Shimizu pointed out that a formulation essentially equivalent to the second variant above appeared earlier as an IMO problem (Combinatorics 5)







The second problem is equivalent to the first problem. There exists a solution to problem 1, if and only if the same sequence of numbers is a solution to problem 2 (note that in problem 2 the counting starts from 0). Therefore we only look at the first and the third problems.

Solution of problem 1:

For small values of n, we can find the solutions by trial and error or with a small program. We denote the numbers of the solution as f(1), f(2), …, f(n).

For n=1, there is a single solution: f(1)=2.

For n=2, there is a single solution: f(1)=4, f(2)=0.

For n=3, there is a single solution: f(1)=4, f(2)=1, f(3)=0.

For n=4, there are two solutions (see the beginning of the post).

For n=5, n=6, there are no solutions.

For n=7, there are two solutions: f(1)=4, f(2)=3, f(3)=0, f(4)=1, f(5)=0, f(6)=0, f(7)=0, and f(1)=5, f(2)=1, f(3)=1, f(4)=1, f(5)=0, f(6)=0, f(7)=0.

For n=8, there are two solutions: f(1)=5, f(2)=2, f(3)=1, f(4)=1, f(5)=0, f(6)=0, f(7)=0, f(8)=0, and f(1)=5, f(2)=3, f(3)=0, f(4)=0, f(5)=1, f(6)=0, f(7)=0, f(8)=0.

For every n>=9 there are exactly three solutions:
a) f(1)=n-3, f(2)=3, f(n-3)=1, and the remaining values 0,
b) f(1)=n-2, f(2)=1, f(4)=1, f(n-4)=1, and the remaining values 0,
c) f(1)=n-3, f(2)=2, f(3)=1, f(n-4)=1, and the remaining values 0.

These solutions clearly work. Let’s prove that no other solutions exist.

We can observe two properties of the sequence f(i):

Property 1: ∑ i * f(i) = 2n. (where i goes from 1 to n)
There are 2n numbers in the sentence and each i*f(i) term counts how many numbers are there with multiplicity i. To have a multiplicity greater than n would mean that all gaps are filled with the same value, which cannot yield a true sentence for any value. Thus the possible multiplicities are between 1 and n, therefore ∑ i * f(i) gives the total number of numbers in the sentence, which is 2n.

Property 2: ∑ f(i) = n+1. (where i goes from 1 to n)
Clearly, for all i, f(i)>=0 and f(i)<=n. If for all i, f(i)>0 would hold, ∑ i * f(i) would be larger than 2n, violating Property 1. Therefore, some of the f(i) values have to be equal to 0, which means that all numbers 0,1,…,n appear in the sentence. Furthermore, all numbers that appear in the sentence have multiplicity between 1 and n. Thus, ∑ f(i) gives the total number of distinct values, which is n+1.

Now we prove that a) b) c) are the only solutions for n>=9.

If f(1)>n-2,
it must be that f(1)=n-1, but now n-1 occurs twice, so to maintain that n-1 values occur once, we need to write n-1 in every other position as well. This is not a correct solution.

If f(1)=n-2,
we must have exactly one other nonzero value (besides n-2) written in the gaps. We cannot write n-2 anywhere else, as that would violate Property 2. To get a sum of n+1, the possibilities are writing one time 3, or three times 1. Writing 3 anywhere doesn’t make the sentence correct, so it remains to write three 1’s. But then the number of 0’s will be n-4, the number of 1’s will be 4 and the number of (n-2)’s will be 2. This gives us solution b).

If f(1)=n-3,
we have two other nonzero values (besides n-3) written in the gaps. They also have to add up to 4 (Property 2). That can be as 3+1 or as 2+1+1. The first case means that 1,3 and n-3 have multiplicities 2 and 0 has multiplicity n-3. The second case means that 2 and n-3 have multiplicity 2, 0 has multiplicity n-4 and 1 has multiplicity 3. These cases uniquely lead to solutions a) and c).

If f(1)<n-3,
suppose f(1)=n-k (where k>3). The value 0 has to appear more than one time (otherwise the sum of Property 1 would be too large). Therefore, among the values 1, …, n, there are k distinct values which are written somewhere in the gaps (to make their multiplicities larger than one in the whole sentence). We know that n-k appears in the first gap, so we have k-1 other values in the gaps starting from the second. Let’s denote these values by a1 < a2 < . . .  < ak − 1. Since they are written in the gaps after the first one, it follows that there are at least a1 + a2 + . . .  + ak − 1 distinct numbers that appear more than once in the sentence. Even if one of them is n-k, and one of them is 0, that still leaves S = a1 + a2 + . . .  + ak − 1 − 2 distinct numbers appearing more than once (therefore, also in the gaps). Since a1 ≥ 1, we have S >= k(k-1)/2 – 2. Even if the smallest of these numbers is 1, we get that f(i) >> n+1, for k>3. This violates Property 2, therefore no solution of this kind is possible.

Solution of problem 3:

For n=1, we have f(1)=1, unique solution.
For n=2, we have f(1)=2, f(2)=0, unique solution.
For n=3, we have f(1)=1, f(2)=1, f(3)=0, unique solution.
For n=4, we have f(1)=2, f(2)=1, f(3)=0, f(4)=0, unique solution.

For n>=5, we have exactly two solutions:
a) f(2)=1, f(n-2)=1, the remaining values 0
b) f(1)=2, f(n-2)=1, the remaining values 0.


Similarly to the first problem, now we have the property:
∑ i * f(i) = n.

Clearly f(n)=0, otherwise all values would have to be equal, which is not a correct solution.
Also, f(n-1)=0. The only alternative would be f(n-1)=1, and all other values equal to zero, but this is not a correct solution either.

If f(n-2)=1 we need that ∑ i * f(i) over all other values is 2. This leads to solutions a) and b) only.
f(n-2)>=2 is not possible, as it would violate the property for n>=5.

Now suppose f(n-k) is the last nonzero value, for k>=3. This means that there are k numbers that are not equal to one of the most frequent values. If 0 is one of the most frequent values, then there are k nonzero values. All of them cannot be equal to 1, as that would not be a correct solution, so ∑ i * f(i) >= k(k-1)/2 + n-k + 1, which is larger than n for k>=3, violating the required property. On the other hand, if less than half of the elements are 0, the property is clearly violated, which shows that there are no other solutions.