What’s the minimum score you could get to win the game? What’s the maximum? Answers here.



Most people I know have played this game. It’s a simple game. It is also addictive. (It’s been clinically proven to be second only to crystal meth… Now that isn’t true of course; my friends from the humanities gave it up quite easily*. I kid, I kid.) Go ahead, give it a go. There are also 3D, 4D, and tetris versions of the game if you care to look for them. The 3D version will be decidedly easier to win than the 2D version, of course. Just harder (and more fun) to visualise. I daren’t start with any of these; I’ve wasted enough time as it is.

The object of the game is to get to the 2048 tile and the “You Win” banner. But I thought I’d put down some perhaps interesting things I worked out about the game.

For instance, you must, if you’ve actually finished the game, know that you can play on past 2048. What do you think is the highest tile you could get to on the 4×4 board?

If you said 65536, like so:


You would be almost correct, unfortunately. You see the game is so designed that both 2s and 4s can be spawned. Which of course means that this can happen:


giving a maximum tile of 2^17, or 131072.

Now, getting your tiles to arrange themselves this way would take more than insane concentration and forethought. It would take the computer conspiring with you to give you tiles in just the right order and at just the right place and so on. The maximum of 2^17 is only a theoretical maximum, then. What the maximum is, when the computer isn’t working to help you, is a much harder thing to calculate. I’ve myself never managed to get a tile higher than 2048, even when things looked promising.

2048_4 2048_3 2048_2 2048


You may have noticed that the game also scores you. The scoring is again relatively simple. For every “merger”, you get as many points as the face value of the tile you just created by merger. Spawned tiles don’t add points. Your score is a measure of how inefficiently you’ve played the game; the ideally played game would leave just one tile of value 2048 and one newly spawned tile on the board.

What then, do you think, is the best (read lowest) score you can theoretically get on your way to a winning game? And what’s the maximum score you can theoretically get as you win the game? This isn’t exactly hard to do, but requires some explanation. The score accumulated as you get to a tile of a certain power of 2 is proportional to both the value of the tile and the exponent. The answer also, of course, depends on whether 2s or 4s are spawned.

Assuming it’s all 2s, for instance, the score will be 8*(3-1) = 16 for a tile of 8. For a tile of 16, the score should be 16*(4-1) = 48. If it’s all 4s that are spawned, the score will be 8 and 32 respectively.

The answers to the maximum question I get are 92164 and 83976 depending on whether exclusively 2s or 4s are spawned. The minimum is either 20480 or 18432 respectively.

My average score for winning games hovers around 23000-25000.  CORRECTION: 23000-25000 is my average score at the end of the game. My average score as I win the game is about 20500. What’s yours?

The one question I would really like to answer but can’t is what happens when a monkey plays the game. This question has proven itself beyond me. Maybe somebody reading can help me out.


*It was pointed out to me that this dig at the humanities is uncalled for. The joke isn’t funny or clever. As stereotypes go, the science-vs-humanities divide is a silly and pernicious one. The innumeracy of people from the humanities is no more a fact than that scientists can’t be good writers or speakers, or any other such similar nonsense. I’ve left in the offending sentence; this may, however, be considered a retraction.

Sydney Brenner on science

Scientific institutions and risk in science.

Sriram sent a bunch of us an interview with Sydney Brenner,one of the discoverers of mRNA. In the broad-ranging interview, Brenner talks about the thrill of making a scientific discovery and the joy of seeing your predictions come true. He talks about how institutions of science are run today and how they might be made better. The entire interview is well worth the read, whether or not you in the end agree with what is being said.

Of interest to me, since this is relevant to what I said about graduate school the last time, is Brenner’s take on taking risks in academic research. In fact, his idea of institutionalising risk, and I’ll say more of why I call it this, reminds me of something that’s said of/to drivers in formula one–somebody who doesn’t once in a while walk back to the pits with the steering wheel in his hands simply isn’t trying hard enough.

Batting for unconventional thinking in science isn’t new. In as much as economics is a science, Paul Krugman’s views on how to do economics better are in the same vein. Brenner says we must stop scientific research from becoming the gigantic machine that makes cogs of graduate students, both by protecting them from financial hardship, and by encouraging independent thought. I wrote the last time about how the Indian research establishment is willy-nilly doing this.

The interviewer also mentions something else that is now disturbingly common: the increasing use of adjunct faculty members, who, even after teaching courses for decades at the same college, are paid little money and have no job security.Several Indian universities do the same thing with adjunct members of teaching faculty as American universities. Government-aided colleges in Karnataka that teach basic science and arts undergraduate colleges haven’t hired permanent members of faculty for something like 25 years. And privately run colleges find it easier to just hire temporary teachers and pay them a pittance.

Brenner also talks fondly of the college system of Oxford and Cambridge and how it allows not only people from different scientific disciplines, but people from different disciplines to sit at the same table. This again is sadly something we haven’t got right. The best institutions in the country focus more and more on smaller and smaller scientific disciplines; science isn’t for us, it seems, to be polluted by mixture with the social sciences or, god forbid, the humanities.

Brenner’s idea of a “Casino Fund” in science — i.e. money that the institutions funding science should allocate for long-shot ideas and basically write it off — is a way of making risk a part of the firmware of science and its institutions. We should allocate a small fraction of all funding and give it to people with a proven record of gambling successfully, he says; this is the only way to ensure that the big risky ideas that might revolutionise science see the light of day. But not everybody agrees. Were the 1950s a golden age for science? A lot of good work was done. But we also know that the 1950s weren’t at all a good time for women in science.

Entirely aware that the extent of my expertise at running an institution of science is to have managed somehow to keep my PhD work afloat, may I venture that perhaps a better implementation of Brenner’s idea of institutionalising risk is to make sure that institutions of science are inclusive? I put it to you that it’s a greater, to say nothing of nobler, risk to take to ensure that science is inclusive. Scientists, being human beings, are products of their time. Could there be anything more limiting than excluding entire categories of people from the humanist endeavour that is science?

Of gay transvestite Irishmen and institutionalised prejudice

I was reading this post on Crooked Timber, that font of proper good sense on all matters economics/sociology. (If you don’t read CT regularly (or at all), you should.)

The post links to the video below, which you might have seen from elsewhere, in which a gay transvestite Irishman, Panti Bliss, makes an important point about institutionalised homophobia and how it can gnaw at even people made of stern stuff. I thought the talk was poignant. I’ve listened to the entire talk a few times now. I can’t do better than Henry Farrell’s analysis, so I’ll just let you read the original post on Crooked Timber.

I would add that what Panti says about homophobia applies more generally to prejudice in society. This is what women go through in misogynistic societies (which is all of them); this is what Dalits in this country face every day; and what black people face in racist societies.

Of depressed graduate students

Notwithstanding what you may have read, PhD students in Indian universities are paid about 4 times the median per capita income in the country (and from what I can make out, about twice the median income in cities, which is perhaps the more relevant figure), and this income is pretty well guaranteed for the duration of our research.

Kavitha pointed me to this The Guardian article about how common mental illness is among PhD students. The article paints a grim picture of what it is to be a graduate student. I am by no means an expert, or perhaps even a typical case; but I do have first hand experience of both mild depression and graduate school (not at the same time, though) and I thought I’d compare my experience with what the article says.

The author of the article is the head of the student counselling unit of some university. The article doesn’t disclose the identity of the author, but I think it’s safe to assume that this is some university in the US.

In short, the author makes the following observations about graduate students from her experience:

1) Many PhD students often work themselves to the point of physical and mental illness, and maybe also think that this isn’t just fait accompli but is also a necessary part of training in academic research.

2) Supervisors are by and large unsympathetic to the plight of graduate students, mostly because people who become professors are usually those people that either did not go through this, or were able to cope.

3) Graduate students and professors view developing a mental illness as an admission of defeat, as a blemish on their career.

They do all this, notes the article almost in passing, while also going through significant financial trouble.

I doubt there are numbers to be found on the prevalence of mental illness in Indian universities, but I do not doubt that students and professors in India are as oblivious to the reality of mental illness as they are in the US. I also have no doubt that there are cases, varying in the degree of severity, of burnout among graduate students with attendant consequences .

In a very important way, however, Indian universities and research centres are different from most universities in the US: notwithstanding what you may have read, PhD students in Indian universities are paid about 4 times the median per capita income in the country (and from what I can make out*, about twice the median income in cities, which is perhaps the more relevant figure). This income is also pretty well guaranteed for the duration of our research, in stark contrast to the constant struggle for funding that defines academic life in universities in the US.

The fact that I have a steady income regardless of what I do with my work makes a world of difference, I think. Is this enough to make the issues the The Guardian article raises irrelevant for Indian universities? I don’t know. I work for professors who have taken all my problems–academic and personal– in stride; but I also know of professors that range from simply uncooperative to downright mean in their treatment of students. It isn’t a stretch to imagine that this could drive a student nuts, and a proper support system would then be necessary and invaluable. My experience has been that the problem cases are few and far between, but I am willing to be corrected on this point.

* How did I get this? The wikipedia page says about a third of India’s population lives in its cities and produces about two thirds the GDP. Even if we assume that all this GDP is equally distributed among the people (and not taken, say, by corporations), this means the average urban dweller makes twice the national per capita income.

Astronomy, evolution, and the tea-seller


There’s this small roadside tea-shop next to the Centre. The man who owns and keeps the shop is from Bihar. We call him Pandey-ji. He’s a colourful man and he is, if not happier, more cheerful than I can be. He is a poet of sorts (do you ‘remember’ Johnny Walker’s sar jo tera chakraye? He writes (not that I know he can write) poetry of that oeuvre). His best one was about this country we live in–his country, our country (mera desh tarakki kar raha hai // har shehar mein chori, daka sab ho raha hai, mera desh tarakki kar raha hai), a little heavy-handed, perhaps, but the sort of anger that I might put down in writing.

Pandey also asks me, every time I go buy chai and smokes from him, something or the other about science. His most frequent questions have been about astronomy and evolution (Hey, is the Earth a sphere, did you say? And we go around the Sun and the Moon goes around us? How high is the Earth’s atmosphere? What’s there outside? Nothing? So a man wouldn’t be able to breathe? How did human life come about, anyway? We evolved from single-celled life, you say? How did that life get here? Panspermia?).

These are not dumb questions by any stretch. I say that because he always adds, after I’ve answered his questions with whatever Hindi I know (what’s Hindi for ‘Oxygen’, do you know? I blabbered for two minutes trying to explain to him what Oxygen is in trying to answer why none of the other planets of the Solar system have life), that he’s an uneducated man who can only ask these questions and try and learn something. These are questions that have bothered philosophers* for millennia; only for the last four centuries have we known that the Earth isn’t the centre of the universe, that it’s a planet in an orbit around the Sun just like half a dozen others. And only for the last century and a half have we known what is today the backbone of all biology, evolution by natural selection. So not dumb questions, then.

My answers, though… they’re a different matter. It isn’t, of course, that the answers are hard. We’ve known the answers for quite some time now, like I said. A reasonable class 12 kid should be able to answer most of these questions. It is that my facility with Hindi can only take me so far (“I can count to nine, and ask where the bathroom is”). What I would really like is to be able to show Pandey the Hindi version of Cosmos (Carl Sagan’s Cosmos, if you insist on being an originalist).

I happen to know Carl Sagan’s Cosmos is available as a book. Is there a Hindi translation/equivalent?

*ADDED IN PROOF: I say “philosophers” here and comment in the next paragraph that class 12 kids could answer most of these questions. I must add that this is not to be read as approval of Steven Pinker’s views on philosophy.