The Surprise Exam Paradox

There are several different versions of this particular paradox – like Holli’s Paradox, and The Unexpected Hanging Paradox. Before I present the surprise examination version, it behooves me to mention that no correct (or final) solution to this paradox has been established yet.

Here’s the paradox:

A teacher announces in class that an examination will be held on some day during the following week, and moreover that the examination will be a surprise.

The students argue that a surprise exam cannot occur. For suppose the exam were on the last day of the week. Then on the previous night, the students would be able to predict that the exam would occur on the following day, and the exam would not be a surprise. So it is impossible for a surprise exam to occur on the last day. But then a surprise exam cannot occur on the penultimate day, either, for in that case the students, knowing that the last day is an impossible day for a surprise exam, would be able to predict on the night before the exam that the exam would occur on the following day. Similarly, the students argue that a surprise exam cannot occur on any other day of the week either.

Confident in this conclusion, they are of course totally surprised when the exam occurs (on Wednesday, say). The announcement is vindicated after all. Where did the students’ reasoning go wrong?

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Similar posts: The Monty Hall Paradox, A Mathematical Conundrum, The Voting Paradox, Broken Clocks.

The Importance of Pedigree

Consider the following thought experiment:

Suppose there is an experience machine that would give you any experience you desired. Superduper neuropsychologists could stimulate your brain so that you would think and feel you were writing a great novel, or making a friend or reading an interesting book. All the time you would be floating in a tank, with electrodes attached to your brain. Should you plug into this machine for life, pre-programming your life’s experience?

This, obviously, is a rhetorical question. And I would assume that for most people answer is a curt “No”. We think that actually playing a game is somehow superior to a mere (stimulated) experience of playing a game; even if the brain attached to electrodes can’t tell the difference. Doing something is better than the experience of doing. We just don’t want to experience things, we want to do thing and be someone.

Here’s another scenario: Person A mistakenly believes that his favorite team won the game last night, and person B correctly believes that his favorite team won. [Assume that there are no consequences of the mistaken belief.] Even if there is no difference in their subjective degrees of happiness, if you had to choose between the two, you would choose to be person B. In terms of well-being of a person, the illusion-based happiness is perceived inferior to the fact-based happiness. You don’t want to rejoice for your home-team just for the sake of being happy. You want your home-team to have actually won as well. Otherwise, it’s just not the same. (Again, even if there’s no subjective difference in what we feel from the inside.)

“Happiness, at least if it is understood to correspond to well-being, turns out not to be all in the head”, writes Leo Katz in his intriguing book Ill-gotten Gains: Evasion, Blackmail, Fraud, and Kindred Puzzles of the Law, from which these scenarios are taken. He quotes an Oxford philosopher to elaborate more on this:

I prefer, in important matters of my life, bitter truth to comfortable illusion. Even if I were surrounded by consummate actors able to give me sweet simulacra of love and affection, I should prefer the relatively bitter diet of their authentic reactions. And I should prefer it not because it would be morally better, or more noble, but because it would make for a better life for me to live.

… because it would make for a better life for me to live. Touché!

The pedigree of experience matters – that’s the lesson from these thought experiments. But alas, when it comes to the belief in God, the majority of us seem to prefer the comfortable illusion — there’s an after-life, God is watching and helping us etc. — to the bitter truth that there’s no God, and we are on our own.

Proof Without Words II

Conjecture: For every natural number n, the sum of the first n odd numbers is n².

In other words:

1 + 3 + 5 + . . . + (2n – 1) = n²

For example, starting with n = 1:

1 = 1²

1 + 3 = 2²

1 + 3 + 5 = 3²

1 + 3 + 5 + 7 = 4²

1 + 3 + 5 + 7 + 9 = 5²

1 + 3 + 5 + 7 + 9 + 11 = 6²

And so on.

This conjecture can be proven by induction, but I think this little picture below does it marvelously and most succinctly:

If you start from the lower left corner, you can see that the first two odd numbers (1 and 3) make a 2 x 2 square. Hence, 1 + 3 = 2². Similarly, the first three odd numbers (1,3, and 5) make a 3 x 3 square. Hence, 1 + 3 + 5 = 3². How cool is that?!

And there’s more: this elegant scheme is also a geometric depiction of the identity (k + 1)² = k² + 2k + 1. Can you spot this proof?

[Source: Excursions in Calculus]

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Also check out an earlier post: Proof without Words.

Santa Venn

Happy Holidays!

Previously on this blog: Lord Shri Santa, And They Have Beards Too! (Another Santa Venn diagram.)

[Hat Tip: Junk Charts]

Knowledge versus Belief

I consider myself an atheist as well as an agnostic. And whenever I mention that to someone, I get a puzzled or disapproving look. (See this comment on one of my previous posts: Religion, Culture and Mr Hitchens.)

Many perceive agnosticism as the middle way between atheism and theism. And people who consider themselves agnostics with that definition in mind seem to feel that both atheism and theism positions are dogmatic. “How can someone be 100% certain that God exists (or does not exist)?”, they ask. From this POV, atheists seem to be making the same mistake as theists (cocksure certitude about an un-provable hypothesis), and agnosticism sounds like a reasonable, open-minded and rational position to take.

This is a flawed understanding of what agnosticism (and atheism) is about. Let’s start with the etymology. The word agnostic comes from the Greek word ‘gnōsis’ which means knowledge. And atheism comes from ‘theo’ which means god. So while agnosticism is about the lack of knowledge about God’s existence, atheism is about the lack of belief in God. They both operate in different paradigms:

In the diagram above,

A = An agnostic theist who does not claim to know for sure whether God exists, but he choose to believe in God anyways.

B = An agnostic atheist who does not believe in God and also thinks that the existence of such entity cannot be known for sure. (This is where I belong. I think the absence of evidence does make God improbable, but not impossible. I do not claim to know for sure that God does not exist. But I think God’s existence is very unlikely.)

C = A gnostic atheist who claims to know for sure that such entity does not exist, and (hence) does not believe in God. This is an unreasonable position – and perhaps very few atheists belong to this category. Atheists usually believe in science and reason, and recognize that all scientific truths are provisional. (“I believe x.” does not mean “I can prove x.”, but “It would be unreasonable to doubt x.” See my earlier post Science Never Proves Anything for more details on this.)

D = A gnostic theist who claims to know that God exists, and (hence) believes in God. I think most theists probably belong in this category, because they are usually pretty sure that their belief is reasonable and justifiable with “evidence”.

So there. Atheism and agnosticism are not mutually exclusive. Also, while some of the above positions are logically more coherent (and hence more prevalent) than others, theoretically, one can belong to any of those four quadrants. I belong to the agnostic atheist quadrant. Where do you belong?

Rockstar: The Music and the Movie

‘Be careful what you ask for, you just might get it.’ – is a one-line, reductive message one can construe from Imtiaz Ali’s latest movie Rockstar. But as I mull over this well-crafted, brilliantly edited movie with an astounding soundtrack (and background score), I realize that this is one of those movies can’t be defined in terms of such meager measures. Like in real life, the joyful and romantic moments pass by in a jiffy, while the sad ones hover around indeterminately. The way the story unfolds in the first half – by the means of snippets, interspersed with flashbacks, and even news clippings – also underscores this point. The masterful craftsmanship by the director Imtiaz Ali is superbly complemented by A R Rahman’s music in this angst-ridden journey of a Rockstar. The story-telling relies and weighs heavily on silent moments, unspoken words, and anguished fury, and often resort to the pulsating soundtrack to express the lead character’s musings, agony and dilemma.

When I first heard that Mohit Chauhan had voiced 9 out of 14 songs in this album, I was skeptical if he would be able to pull it off. But as I heard ‘Phir Se Ud Chala’ (few months ago when the album released), the choice of Chauhan started to make sense. He hails from the mountainous regions of Himachal Pradesh and has mastered the local pahadi folk music style. His reverie-inducing, dreamy, almost careless rendition is exactly what this song needed. The song has a very effervescent quality. Nothing repeats itself, but there’s an underlying order in apparent randomness. The beginning chorus reminded me of the ‘Sone ka pitara’ line from the ‘Jahan Piya’ [Pardes] – which is probably influenced by a traditional wedding song in Northern India.

Imtiaz Ali effectively utilizes most of the 14 tracks. The only song that doesn’t get materialized fully on screen is ‘Tum Ko’ (the Kavita Krishnamurthy version). Harshdeep Kaur’s playful ‘Kateya Karoon’ plays in the background and conveys Heer’s adolescent and naughty ambitions well. I kept thinking that ‘Kun Fayakun’ was somewhat redundant – if not forced into the script. It doesn’t really add anything to the story. (Except for the fact that Shammi Kapoor notices him at the dargah, but that could have happened anywhere.) The whole spiritual awakening was an unnecessary detour that was irrelevant to Jordon’s transformation into an artist. The actual transformation was triggered by his broken heart and lost love; the Sufi spiritualism seemed out of place in Jordan’s life given his character.

‘Jo Bhi Mein’ is so melodious and instantly catchy that it’s hard to believe that Irshad Kamil wrote the lyrics of this song first and ARR developed the tune afterwards. I just loved the guitar work (by Kabuli and Shon Pinto) complimenting the simple tune, and Kamil’s commendable poetry. This song is picturized during the early phases of Jordon’s musical journey – before the real artist is born. Hence the undemanding/easy tune makes sense. On the other hand, the anthem-like, guitar-laden, stunning ‘Sadda Haq’ comes much later as he matures as a musician. The contrast in Ranbir’s expressions between these two songs is enough for one to realize how much hard work he has put in to portray this character. The way Ranbir has performed this song, the tensed nerves on his neck, fuming mouth shouting into the microphone, moody head swings, and angry eyes full of chutzpah make you believe that you are looking at a true Rockstar who really knows the game. Notice how he hastily shakes his fingers several times before starting to play guitar. These small yet careful touches by Ali and Ranbir make this movie a treat to watch.

‘Sheher Mein’ was such a perfectly psychedelic pastiche (ARR doing Nadeem Shravan!) that I was a bit disappointed that it was reduced to few lines in the movie. All the interjections (‘Tum sun nahi rahe ho yaar’ etc.) were retained though, and serve well as comic relief. This is a fun song, something ARR has rarely done before. Other amusing song in this album is ‘Hava Hava’ with a complicated structure where ARR takes some inspirations from gypsy rhythm and sound. Chauhan does Masakkali again, and his voice suits this song well. I especially liked the lyrics, a Raja-Rani anecdote that also reflects (in parts) Heer’s dilemma as she trudges along her unhappy marriage.

Before watching the movie, I felt that the fireworks of acoustic and electric guitar riffs and heavy drum beats (by Shivmani) in ‘Naadaan Parindey’ tried a little too hard to take this song to a climactic height that it wasn’t really meant for. I felt that the extraneous orgy of sounds in this song was borderline dissonant at times. But this song was metamorphosed very successfully on screen – again, kudos to Ali and Ranbir. This is the only song that breaks the one-singer rule as some of Ranbir’s lines were rendered by ARR. In spite of ARR’s presence in the elevated mukhda, I think the song really gets life when Mohit enters with ‘Kaate chahe kitna…’ and ‘Kaga re kaga re…’ lines.

Surprisingly for a love story, there are no duets in this movie. And ‘Tum Ho’ is the only song that Jordon actually sings to Heer. (All other songs are either in the background, or are performed by Jordon to some general audience.) And I thought this was a wonderful way to end this movie – it shows snippets from the times they spend together. Not only we see Jordan sing for and to the object of his inspiration, but we are also reminded yet again how short-lived their romance was; how little time they actually spent together. (A side note: Their adolescent love affair started as a gimmick which involved list-making of things that are social taboo. Eventually, as their relationship matured, they end up breaking one of the biggest social customs – their illicit love affair carries on in complete disregard of the institution of marriage. How ironic!)

Finally, the cinematic pinnacle of the movie for me was the picturization of ‘Aur Ho’. Not only did ARR put Chauhan’s singing ranges on anvil (and Chauhan scored triumphantly), but Ali’s direction, Anil Mehta’s cinematography, Kamil’s lyrics, and the display of emotions/dilemma by the lead actors also attain an artistic high in this song. Some of the best lines in the album come from this song – like ‘Tujhe pehli baar mein milta hoon, har dafa’ and ‘Mein hasrat me ek uljhi dor hua…’. And what can I say about the way this song ends – that unruly kiss at the end of a passionate performance! Loved it! Imtiaz Ali, take a bow.

[Picture Courtesy: The official movie site of Rockstar]

Proof Without Words

A couple of months ago, Google honored one of the most famous problems in mathematics — known as Fermat’s Last Theorem — on the 410th birth anniversary of the man who came up with the theorem: Pierre De Fermat. Here’s the Goodle doodle from August 17th, 2011:

The Fermat’s Last Theorem is really a higher-dimensional extension of the Pythagoras theorem that operates in 2 dimensions. Do you remember the proof (or, a proof) of the Pythagorean theorem? Here’s one:

Take a look at the right angle triangle shown below, the sides are: a=3, b=4 and c=5. Every side of the triangle is expanded to create squares A, B and C. If we slice square A into smaller squares of 1×1, we get 9 small squares. Similarly, by cutting square B we get 16 small squares. In total, we have 9+16=25 small squares — which can fit into square C which needs 25 1×1 squares.

In other words, the area of square C is equal to the combined areas of A and B. Hence, a² + b² = c²! This is a visual proof of the Pythagoras theorem — a proof without words.

And Fermat’s Last Theorem is a projection of this into a higher dimensional space. He hypothesized that it is not possible to do this kind of perfect-fit re-arrangement for objects that have more than two dimensions. For example, instead of making squares by projecting the sides of the triangle, if you make cubes (i.e. 3 dimensions instead of 2) then you can’t cut those smaller cubes (A and B) in any way that can give you enough to fill the larger cube. Dissecting A and B into cubes of 1x1x1 would give us 27+64=91 small cubies, but the larger cube needs 125 to be completely filled.

Hence, aⁿ + bⁿ is not equal to cⁿ when n>2. And that’s known as Fermat’s Last Theorem.

Fermat himself left no proof of his conjecture, and it remained one of the most famous unsolved problems in mathematics. It took around 350 years for mathematicians to prove Fermat’s Last Theorem! The proof involves some heavy mathematics, and beyond the scope of this post (as well as my limited mathematical knowledge.)

What intrigues me though is the idea of ‘proof without words’ (like a layman’s proof I explained above.) For Pythagorean theorem there exists numerous such proofs that require not even a single word to explain. Here are few of my favorite examples of such visual proofs:

I think the following two are really splendid and quite elegant:

Click to embiggen – it may take some time to spot the proofs in these patterns. The second proof by dissection above is actually ascribed to Bhaskara, the great Indian mathematician from the 12th century.

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Image courtesy: The first demonstration (proof by rearrangement) is from the Wikipedia link on Pythagorean Theorem, and the other two (proofs by dissection) are from this paper. There are many more such proofs on this wonderful page.

A 48-minute documentary about the mathematician who solved Fermat’s Last Theorem is available here.

On Punctuation

This little passage from a Wikipedia article made me chuckle:

Parentheses may also be nested (generally with one set (such as this) inside another set). This is not commonly used in formal writing (though sometimes other brackets [especially parentheses] will be used for one or more inner set of parentheses [in other words, secondary {or even tertiary} phrases can be found within the main parenthetical sentence]).

The grammatical rules regarding punctuations and parentheses are not very precise. Hence, one can easily construct a paragraph that’s grammatically, syntactically, and logically accurate, but absurd at the same time. Sometimes the results can be quite amusing (as in the excerpt mentioned above.)

Here’s another example where the author deliberately uses numerous parentheses to explain one thing: that there are no precise rules about punctuation:

There are no precise rules about punctuation (Fowler lays out some general advice (as best he can under the complex circumstances of English prose (he points out, for example, that we possess only four stops (the comma, the semicolon, the colon and the period (the question mark and exclamation point are not, strictly speaking, stops; they are indicators of tone (oddly enough, the Greeks employed the semicolon for their question mark (it produces a strange sensation to read a Greek sentence which is a straightforward question: Why weepest thou; (instead of Why weepest thou? (and, of course, there are parentheses (which are surely a kind of punctuation making this whole matter much more complicated by having to count up the left-handed parentheses in order to be sure of closing with the right number (but if the parentheses were left out, with nothing to work with but the stops we would have considerably more flexibility in the deploying of layers of meaning than if we tried to separate all the clauses by physical barriers (and in the latter case, while we might have more precision and exactitude for our meaning, we would lose the essential flavor of language, which is its wonderful ambiguity)))))))))))).

And while we are on the topic, let me mention two of my pet peeves regarding the usage of (1) writing in CAPS, and (2) misuse of quotes:

(1) Every once in a while I come across a sentence (sometimes an entire email) written in all caps. Writing in all caps doesn’t imply importance, seriousness, or emphasis – and they don’t help make your proposition more convincing (as the writers of these horrid sentences seem to think.) Italics should be used for emphasis; writing a whole sentence in all caps is equivalent to shouting. Please don’t do that (unless it’s warranted.)

(2) The unnecessary and erroneous usage of quotation marks. Again, some people seem to think that quotation marks imply emphasis. They don’t. Quotes should be used to indicate direct speech, different meaning of a word, and sometimes irony. Here’s one such misuse of quotation marks:

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By the way, did you know that Sanskrit originally did not have any punctuation marks? The vertical bar (|) and double vertical bars were added later in the 17th century. Could that be one of the many reasons behind its eventual demise as a vernacular language in India?

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The article mentioned above is available here, and was written by Lewis Thomas. Oh, and I share the author’s fondness for semicolons.

Hat tip for the Wikipedia extract goes to xkcd, and the billboard image is taken from The “Blog” of “Unnecessary” Quotation Marks. Yes, there’s an entire blog dedicated to unnecessary quotes.

What I’ve been Reading

(1) Here’s Looking at Euclid was a delightful read. Alex Bellos takes the reader on an exciting journey of mathematical curiosities. The subtitle says it all: From Counting Ants to Game of Chance – An Awe-Inspiring Journey Through the World of Numbers. The chapters are written – in a relaxed prose style – as stand-alone articles, which was really convenient for me as my reading schedule is becoming quite intermittent lately. This recreational math book is for you if any of the following gets you excited: abacus, Sudoku, Pythagoras theorem, convergence of infinite series, approximations of pi, duodecimal system and such. (A minor gripe: I thought the pun in the book-title was quite silly.)

(2) I picked up At Home: A Short History of Private Life a couple of months ago after reading an excerpt that tickled my fancy. The book was supposed to be about questions like ‘Why did we end up with salt and pepper on our dining tables, rather than say, salt and cardamom?’, ‘Why do we have four tines in forks’? The components and features of our homes are so familiar that we don’t pay much attention to them. The history of domestic life is an unexplored territory, and I thought it can be very interesting. But as I read through the initial chapters replete with British social history and anecdotes, my excitement quickly turned into apathy. The book is enjoyable in parts, but mostly it was a tedious ramble about the history of English country houses.

(3) I am halfway through The Disappearing Spoon, and it has been an interesting read so far. In a way, this book is like a memoir of elements in the Periodic table. The first chapter starts with a novel introduction to the Periodic table — an “anthropological marvel” in Sam Kean’s opinion. He takes each element in the periodic table – mercury, thallium, silicon, carbon, hydrogen, etc. – and tells engaging stories associated with these elements that were so foreign to me and at first reminded me of my befuddlement when my high-school teacher introduced this chart to our class years ago. Reading through these funny, engrossing, and often chilling episodes, I am getting acquainted with these elements, their properties and idiosyncrasies without even realizing that I am learning science. I never thought that periodic table can be this much fun. (I did like the practical joke in this book’s title – a disappearing spoon made of gallium.)

Fibonacci Spiral

The Fibonacci series emerges again, and this time its wearing quite an impending and ominous cloak: the hurricane Irene. Here’s a satellite image of the hurricane looming over the Eastern coast of US – with Fibonacci spiral overlay.

Note that the eye of the hurricane is a bit off-center – which calls for a disclaimer. It’s not difficult to find an approximation of any mathematical relation in nature which is replete with myriad shapes. There’s no divine, mystical governing principle that guides the shape and size of objects.

Having said that, there are situations in which a Fibonacci  (and other) series may arise naturally. See my earlier post Nature by Numbers for some interesting examples.

[The image is taken from here.]