Tag: education

Muggoos and overfitting

Back when I was a student, there was this (rather large) species of students who we used to call “muggoos”. They were called that because they would have a habit of “mugging up the answers” – basically they would learn verbatim stuff in the textbooks and other reading material, and then just spit it out during the exams.

They were incredibly hardworking, of course – since the volume of stuff to mug was immense – and they would make up for their general lack of understanding of the concepts with their massive memories and rote learning.

On average, they did rather well – with all that mugging, the downside was floored. However, they would stumble badly in case of any “open book exams” (where we would be allowed to carry textbooks into the exams) – since the value of mugging there was severely limited. I remember having an argument once with some topper-type muggoos (with generally much better grades than me ) on whether to keep exams in a particular course open book or closed book. They all wanted closed book of course.

This morning, I happened to remember this species while chatting with a friend. He was sending me some screenshots from ChatGPT and was marvelling at something which it supposedly made up (I remembered it as a popular meme from 4-5 years back). I immediately responded that ChatGPT was simply “overfitting” in this case.

Since this was a rather popular online meme, and a lot of tweets would have been part of ChatGPT’s training data, coming up with this “meme-y joke” was basically the algorithm remembering this exact pattern that occurred multiple times in the training set. There was no need to intuit or interpolate or hallucinate – the number of occurrences in the training set meant this was an “obvious joke”.

In that sense, muggoos are like badly trained pieces of artificial intelligence (well, I might argue that their intelligence IS artificial) – they haven’t learnt the concepts, so they are unable to be creative or hallucinate. However, they have been “trained” very very well on the stuff that is there in the textbooks (and other reading material) – and the moment they see part of that it’s easy for them to “complete the sentences”. So when questions in the exams come straight out of the reading materials (as they do in a LOT of indian universities and school boards) they find it easy to answer.

However, when tested on “concepts”, they now need to intuit – and infer based on their understanding. In that sense, they are like badly trained machine learning models.

One of the biggest pitfalls in machine learning is “overfitting” – where you build a model that is so optimised to the training data that it learns quirks of the data that you don’t want it to learn. It performs superbly on the training dataset. Now, when faced with an unknown (“out of syllabus”) test set, it underperforms like crazy. In machine learning, we use techniques such as cross validation to make sure algorithms don’t overfit.

That, however, is not how the conventional Indian education system trains you – throughout most of the education, you find that the “test set” is a subset of the “training set” (questions in examinations come straight out of the textbook). Consequently, people with the ability to mug find that it is a winning strategy to just “overfit” and learn the textbooks verbatim – the likelihood of being caught out by unseen test data is minimal.

And then IF they get out into the real world, they find that a lot of the “test data” is unknown, and having not learnt to truly learn from the data, they struggle.

PS: Overfitting is not the only way machine learning systems misbehave. Sometimes they end up learning the entirely wrong pattern!

Business school all over again

This morning, I felt like I was in business school all over again.

So the Montessori school that my daughter goes to is exploring the possibility of introducing an adolescent (12-18 age group) program that follows the Montessori philosophy. Towards this end, they are having a series of “seminars” with parents to explain the methodology and collect feedback.

Before the first such “seminar” two months ago, they had sent us all a paper written by Dr. Maria Montessori and asked us to read it in preparation. When we walked in to school, we were all given copies of the same paper and asked to read it before the discussions started. The teachers walked in after having given all of us to read through the paper once again.  “This sounds like Amazon”, I had thought.

To give parents full flavour of the proposed program, we were told that these sessions mirror what the adolescent version of the school is supposed to be like. Each session involves discussion of a piece of written text. All participants are supposed to have read it beforehand. And discussions have to be on point to the reading – like every note of participation has to refer to a particular page and paragraph. I had come away from the first session thinking “these guys seem to be trying to recreate business school in high school”.

And then, this morning, at the second such session, I got a taste of this medicine as well. I’ve had two insanely productive days at work last two days, which has meant that evenings I’ve been rather tired and unable to really read the paper (once again it was a paper by Dr. Montessori). This morning, I woke up late and by the time I got to school for the session (that began at 8am), I’d barely managed to glance through the paper.

I furiously tried to read it before the teachers came in, and barely managed a fourth. The teachers reminded us of the rules – all discussing points had to refer to specific parts of the paper, and we couldn’t talk “generally” (ruling out any “arbit class participation”). Also, the teachers would not “lead” the discussion – the format of the class was such that it was peer discussion.

I’m speculating here, but it is possible that many other parents this morning were also in my state – having turned up to class having not read the prescribed reading. Initially the CP was slow and deliberate. That we had to reply to each other (and keep referring to the text as we did so) made it slower. There were a few awkward pauses which I tried to use to hurriedly read the rest of the paper. I was also getting distracted, planning this blogpost in my head. I was also simultaneously feeling horrible about not having come to the session prepared, and was thinking I’m a horrible parent.

The format of the discussion helped, though, as different people kept referring to different sections of the paper, and I sort of read through it in a non-linear fashion. In about ten minutes, in the course of the discussion I had probably read through the entire text. And then I started unleashing.

All those business school skills came of good use – despite the constrained format, I somehow winged through today’s session (not that that was the intended consequence). By the end of the session I had comfortably spoken the most in the group. Old habits die hard, I guess.

It weirdly felt like I was business school once again. And as it happened, I noticed that the person next to me was wearing an IIMB T-shirt (though he didn’t put too much CP)!

On a more serious note, maybe this kind of a schooling format in high school might mean that the children may not really need to go to college!

Diversity and campus placements

I graduated from IIMB in 2006. As was a sort of habit around that time in all IIMs, many recruiters who were supposed to come to campus for recruitment in the third or fourth slot were asked to not turn up – everyone who was in the market for a job had been placed by then.

The situation was very different when my wife was graduating from IESE Business School in 2016. There, barring consulting firms and a handful of other firms, campus placements was nonexistent.

Given the diversity of her class (the 200 odd students came from 60 different countries, and had vastly different experience), it didn’t make sense for a recruiter to come to campus. The ones that turned up like the McKinseys and Amazons of the world were looking for “generic management talent”, or to put it less charitably, “perfectly replaceable people”.

When companies were looking for perfectly replaceable people, background and experience didn’t matter that much. What mattered was the candidate’s aptitude for the job at hand, which was tested in a series of gruelling interviews.

However, when the jobs were a tad more specialised, a highly diverse campus population didn’t help. The specialisation in the job would mean that the recruiters would have a very strong preference for certain people in the class rather than others, and the risk of not getting the most preferred candidates was high. For specialised recruiters to turn up to campus, it was all or nothing, since the people in the class were so unlike one another.

People in the class were so unlike one another for good reason, and by design – they would be able to add significantly better value to one another in class by dint of their varied experience. When it came to placements, however, it was a problem.

My IIMB class was hardly diverse. Some 130 out of 180 of us were engineers, if I remember correctly. More than a 100 of us had a year or less of real work experience. About 150 out of 180 were male. Whatever dimension you looked at us from, there was little to differentiate us. We were a homogeneous block. That also meant that in class, we had little to add to each other (apart from wisecracks and “challenges”).

This, however, worked out beautifully when it came to us getting jobs. Because we were so similar to one another, for a recruiter coming in, it didn’t really matter which of us joined them. While every recruiter might have come in with a shortlist of highly preferred candidates, not getting people from this shortlist wouldn’t have hurt them as much – whoever else they got was not very dissimilar to the ones in their original shortlist.

This also meant that the arbitrarily short interviews (firms had to make a decision after two or three interviews that together lasted an hour) didn’t matter that much. Yes, it was a highly random process that I came to hate from both sides (interviewee and interviewer), but in the larger scheme of things, thanks to the lack of diversity, it didn’t matter to the interviewer.

And so with the students being more or less commoditised, the incentive for a recruiter to come and recruit was greater. And so they came in droves, and in at least my batch and the next, several of them had to be requested to not come since “everyone was already placed” (after that came to Global Financial Crisis, so I don’t know how things were).

Batch sizes at IIM have increased and diversity, too, on some counts (there are more women now). However, at a larger level I still think IIM classes are homogeneous enough to attract campus recruiters. I don’t know what the situation this year is with the pandemic, but I would be surprised if placements in the last few years was anything short of stellar.

So this is a tradeoffs that business schools (and other schools) need to deal with – the more diverse the class, the richer will be the peer learning, but lesser the incentive for campus recruitment.

Of late I’ve got into this habit of throwing ideas randomly at twitter, and then expanding them into blog posts. This is one of those posts. While this post has been brewing for five years now (ever since my wife started her placement process at IESE), the immediate trigger was some discussion on twitter regarding liberal arts courses.

 

The Puritan Topper

This was an idea that sort of got ingrained in my head at the turn of the millennium – around the time I was transitioning from school to undergrad. That you would be a topper if and only if you led an otherwise diligent and disciplined life.

For starters you needed to be a nice person (among the things this entailed for a potential topper was to liberally share notes and clarify people’s doubts when called upon). You weren’t allowed to have any character flaws. You weren’t supposed to get distracted with things like hitting on someone or being in a romantic relationship. You would talk to, and be polite with, people of the opposite sex, but “nothing more than necessary”. “Bad habits” like smoking and drinking were out of the question.

These were just the necessary conditions. On top of this, of course, you had to work with single-minded devotion towards becoming the topper. You needed to be diligent, be rigorous with all your assignments, study more than anyone else and all that.

I don’t know how this view of the “puritan topper” got formed in my head. Maybe it was pattern recognition based on the profile of people who used to top in my schools (this was after I had all but given up on doing well academically, apart from entrance exams), especially in undergrad.

I’m also wondering if this image of the puritan topper had something to do with my own giving up – while I might have had the enthu to work hard at academics and do well, this sort of a puritan lifestyle that I had come to associate with toppers (I didn’t smoke or drink, but being nice to everyone all the time was well beyond me) seemed rather daunting.

In any case, this image of the puritan topper didn’t last long. At IIMB, for example, there was this guy who lived a few doors away from me who spent most of his time drinking and hardly any time studying, but aced all exams. Another guy quickly found himself a girlfriend, but continued to top. Suddenly, I found that “normal people” could be toppers as well, and that my view of the puritan topper had been formed mainly on a small number of data points and didn’t hold.

Yet, the number of years that this puritan topper image stayed in my head means that it’s one that has been hard to shake off. A couple of years back, for example, the all india topper in the IIT-JEE, while talking to the press, expressed tribute to his girlfriend for her support. While it’s normal for a class 12 person to have a girlfriend, this comment sort of threw me off – it didn’t fit my mental image of the puritan topper.

Sometimes it is possible to form an irrational belief based on a small number of data points, and irrespective of the number of data points you see to the contrary, it becomes hard to let go of these beliefs. And that makes you more irrational. But I guess, there’s no logic to a lot of these beliefs. Maybe as Rory Sutherland puts it, it’s all “psycho-logic”.

Teacher abuse

Historically, it has been acceptable, indeed desirable, for the teacher to abuse students. Our epics are full of stories where the teacher plays elusive, challenging students to “prove themselves worthy” before being imparted learnings.

The most famous example, of course, comes from the Hindu myth story of Ekalavya who gave a finger to his non-teaching Guru Dronacharya. Elsewhere in the Mahabharata, we had Parashurama cursing his student Karna after discovering that the latter was not a Brahmin.

It is not just Hindu mythology that has such stories (just that I’m most familiar with this). In Quentin Tarantino’s Kill Bill, for example, Pai Mei abuses his pupils, making them carry water up the hill and serve him otherwise until he teaches them the five point exploding heart technique. He drives his students to such a rage that one of them (Elle Driver) ends up killing him.

And this privileged attitude of the teacher (“acharya devo bhava“) extends to modern universities as well. It is common for advisors to endlessly push graduate students before they permit them to graduate, or to take credit for graduate students’ work (check out PhD comics.). In IIT Madras, where I did my undergrad, it is reportedly common for professors to endlessly flunk students who have pissed them off (I played it safe, so no first hand experience in this). Schoolteachers hand out corporal punishment, which is only recently making its exit from the classroom.

As part of my portfolio life over the last seven years, I’ve done several teaching jobs. I’ve taught at IIM Bangalore as an Adjunct Professor. I’ve conducted Data Journalism workshops for journalists and PR executives. I’ve done corporate training workshops.

In the initial days, I would sometimes act like a “typical teacher”, getting annoyed with students with this or that, or abusing my position of privilege in the classroom. Over time, though, I’ve come to see my students as clients – after all, they’re paying me (directly or indirectly) to teach them. And I’ve come to understand that they need to be treated like I treat my other clients – with respect.

If the fact that students are teachers’ clients is this intuitive, why is it that teachers everywhere (both in history and contemporarily) have found it acceptable to abuse students? Is it because teachers are sometimes able to hide behind the brands of sought-after schools and universities? Is it due to the concept of tenure, where professors are recruited for research prowess, and student feedback doesn’t really matter?

Or is it just a self-fulfilling prophecy? Once upon a time, teachers were scarce, and could hence put up their price, and chose to extract it not in cash but in other means. And so the image of “teacher is god” got formed, and perpetuated since most students decided to adhere to it (at least when the teacher is around). To add to this, over time we’ve created institutions such as university rankings which continue to push up artificial scarcity of teachers.

Do you have any idea on why teachers abuse their clients?

Conversation with an Afghan-Dutch taxi driver

We got back to London yesterday, and were welcomed with atypical London weather – thunderstorms. While it is common to stereotype London’s weather as being typically shitty and grey, it doesn’t normally rain all that heavily here – most of the rain that London gets is what is called “spitting rain” – slow drizzly rain best dealt with with a nice cap.

Also welcoming us was an Afghan-Dutch guy who drove us home in his Merc (we hired him through Uber). We got talking and there were a few interesting things from what he said that I though were Pertinent.

  • When we told him we were from Bangalore he said something that sounded like “cooley”. First we interpreted it as him saying that the city is cool, and then realised that wasn’t what he was saying. Then I thought he was talking about Coolie which was filmed in Bangalore, but it wasn’t that as well. Finally we realised he was talking about Virat Kohli, who plays for Royal Challengers Bangalore. It’s funny how Kohli is identified with Bangalore abroad though he’s only nominally based there only during the IPL season
  • We spoke a bit about the IPL and he said he was disappointed that “our team” lost. A minute later he said the team was Sunrisers Hyderabad. For a while it wasn’t clear as to why the Sunrisers were his team. Then I realised they have two prominent Afghan players – Rashid Khan and Mohammad Nabi.
  • He was studying to be a dentist, and decided to spend time in England learning English because a lot of the dental course was in English. Apart from putting himself through formal English classes, driving an Uber was a way for him to become better at English (it’s interesting how at times in our conversation he switched to using Hindi words – some of which I’m guessing are common to Pashto as well), apart from making money
  • My wife later told me that it was common for continental Europeans to spend a gap year in England learning English. And that apart from taking classes they take up jobs where they can practice the language – like driving a taxi or waiting tables.
  • The conversation also got me thinking about gap years and saving up for education – something that doesn’t at all happen in India. In India, the standard practice is to go to college immediately after school, when one is still being funded by parents. In one way, this reduces social mobility since people whose parents can’t afford college end up not studying. Also, the returns to education in India are high enough that the compensation for blue collar jobs (that one can find without a college degree) isn’t enough to fund a later degree.
  • Despite having Afghan parents, this guy has never been there. “It’s way too dangerous. I can go see relatives but will end up spending most time indoors, so not much fun”, he said.

Every time I have a conversation with a taxi driver I’m reminded of what I was told by a friend on the day I moved to Delhi in 2008. “It might be common in Bangalore to chat up auto and taxi drivers”, he had told me, “but in Delhi it is not the done thing”. I still wonder why.

1/13: Leaving home

Tomorrow, Pinky turns 30. I set out wanting to write 30 blogposts about her on the occasion. As it has happened, I managed 13 before I ran out of ideas and time. Anyway, I hope she likes them! 

Sometimes it’s hard to understand what some people are going through. When they put up a brave face and tell you that everything is okay, and they don’t crib, you simply assume that all is right with them. You don’t once try to understand that there might be some struggles going on within, and that the brave face is a result of being able to somehow deal with all of that.

Pinky hasn’t had the last three years easy. In August 2014, she moved to Barcelona to live by herself for the first time ever (she used to live with her parents until she moved in with me in 2010). The small matters of living alone for the first time, and in a new country, were compounded by lack of funds. We’d purchased an apartment in Bangalore earlier that year, and had exhausted a lot of our savings for that.

Unsure of how much she had to spend, Pinky economised. She would write a long email to me every day (and I’d wake up every morning looking forward to that mail), and while she seemed to be having a good time meeting new people and partying late into the night (on many days I’d be awake in Bangalore by the time she got home in Barcelona), she was also careful about conserving money.

There were times when she’d go out with new-found friends and not eat anything because the restaurant was too expensive. She’d ask for tap water, or the cheapest drinks, on nights out so that she didn’t blow away the savings. For breakfast she had buns and croissants bought in bulk at supermarkets – that came at a big discount.

She told me she looked forward to my visits to Barcelona in the hope that she could “spend normally”. In her last term when I lived with her in Barcelona, our monthly spending was three times what she normally spent when living alone!

And Barcelona was hardly the toughest part of her MBA. Her focus on e-commerce and operations had taken her for an internship to Jakarta, where she landed right in the middle of Ramzan. With her office being in an out-of-the-way warehouse, there were no lunch options available nearby, and she spent nearly the entire month without lunch, going all day hungry. Also a delay in her pay and reimbursement had led to a working capital crunch, which nearly left her homeless (it ultimately didn’t get THAT bad).

It was similar later that year when she was in U. Michigan as an exchange student. She survived an entire term without a lamp in her room (it was an unfurnished house), and slept on the floor on a mattress another student had donated to her. Food was also a struggle, as being the only woman among a bunch of Indians left her as the “resident cook” of her apartment. And the US sprawl meant she couldn’t get nutritious ingredients, which were only available at far-off supermarkets.

Yet, whenever we spoke, she was mostly positive and seldom cried. Irrespective of the difficulties she went through, she was focussed on her academics and career. It was only much later, after she had graduated that she had told me how she’d gone through really tough times.

And even amidst the toughness, she remained resourceful. She found that her US Visa allowed her to work on campus, and managed to make some money as a teaching assistant. Back “home” in Barcelona, she wrote cases and made more money. And despite some setbacks, she kept her job-hunt going, graduating with a much sought-after job with Amazon.

I’m proud to be married to her! And you might wonder why I’m suddenly writing all this – she turns 30 tomorrow, and this is as good a time as ever to express my gratitude to her!

Batch size at IIMB

A few days back I had written about how the new IIMs with a sanctioned batch size of around 60 and a faculty strength of 20 are unviable and need to scale up quickly. My argument was that one of the big strengths of the older IIMs is its faculty size which leads to a large number of electives, which allows students to shape themselves the way they best feel. In this context it would be interesting to compare these IIMs to one or more of the older IIMs.

I recently received a mail by the IIMB Alumni Association asking me to reach out to batchmates who are not part of the association. This mail had been sent to all IIMB Alumni who are registered with the association, and the purpose was to increase membership and reach of the association (and no, there are no membership fees). And the mail came with a very interesting data set, and one of the fields was the size of each graduating batch at IIMB.

Source: IIMB Alumni Association
Source: IIMB Alumni Association

It can be seen that IIMB also started rather small, with about 50 students graduating in the first batch in 1976. By the end of the decade, the number was close to a 100, which is where it stayed through the 1980s. Around 1990 was when the batch size increased to about 150, and the number stayed within the 150-200 range for another decade and a half (the 2004 batch was bigger than the ones around it, possibly due to the IT slowdown in 2002 when this batch entered IIM).

And then after 2006 (when I graduated), the batch size increased. My batch had three sections as would have the 15 batches prior to that (based on this data; IIM sections normally consist of 60-70 students). In fact, the “quantum” nature of the increase in batch size at IIM can be put down to the concept of sections – so the increase from the 100 to 150 level was a function of addition of a third section, and so on. After 2006, though, the batch size has exploded, and the current batch (2013-15, who I’m teaching) has a strength of almost 400 students (divided into six sections).

A good addition to this dataset would be some data that could show the prominence or measure of success of IIMB Alumni who graduated in each  batch, which can then allow us to examine whether batch size has had anything to do with continued career success of the students. It would be interesting to examine how this additional data can be collected.

Why Cash Transfers Should Not Replace Midday Meals in Schools

Admin Note: This is not a typical RQ post, in that this has no numbers. Yet, since this is policy related I think it makes sense to put it here

I’m normally a big fan of cash transfers. I’m glad that the Indian government has started implementing it for things like fuel subsidies and certain other benefits. After all, by simply providing the subsidy in cash (market price minus intended “subsidized price”) the government achieves the subsidy while not really having to bother about managing the supply chain. I would have been less unhappy with the Food Security Bill had it been designed as a cash transfer scheme, rather than giving further responsibility to the much-maligned Public Distribution System. With the midday meal system in schools, though, I make an exception.

Following the tragedy in the Bihar school this week, people have called for the government to scrap the midday meal scheme in schools and provide students a cash subsidy instead. Some people have argued against it quoting economies of scale (for example, ISKCON, under its Akshaya Patra scheme, provides midday meals to children in Bangalore schools at the cost of Rs. 6 per child per day, and that amount cannot but much food in the market). That aside, there is a fundamental economic argument against providing for children’s midday meal in the form of cash.

Every year, during Christmas time, journalist Tim Harford (of Financial Times, BBC Radio 4, etc.) writes an article that states that gifting induces a net weight loss, and the economically ideal way of gifting is to gift cash. For example, if I give you Rs. 100 in cash, you can do whatever you want with that cash. Instead, let us say that I use the Rs. 100 to buy you a gift (let’s say a pen). Now, irrespective of how much value you see in the pen, you don’t have the option any more of spending that Rs. 100 on anything except the pen I’ve got you. So you are in effect poorer than you would be had I simply given you the Rs. 100 rather than buying you the pen.

The question now is whether I want you to buy more pens or less. If for some reason I believe that buying more pens is good for your health, I can do my bit in encouraging that behaviour by gifting you pens rather than gifting cash. If on the other hand I don’t have a view on whether pens are good for you, I will gift you cash.

Government subsidies work the same way. If the government wants to encourage consumption of a particular good or service, it subsidizes it directly. If, on the other hand, the government doesn’t have a particularly strong view on whether a citizen should consume more or less of a particular good, but only wants the citizen to be able to afford that particular good, it provides cash. With the current form of the food security bill (where the government has promised to give foodgrain at subsidized prices) the government is implicitly stating that it wants to encourage people to eat more foodgrain (which flies in the face of data which shows that most Indians already eat too much cereal and too little of other nutritious foods). If the intention were only to ensure that people can afford food grains, a cash transfer would have sufficed. Similarly, by moving to a cash transfer scheme for cooking fuel, the government has signaled that it doesn’t particularly encourage the use of cooking fuel, but it simply wants to make it affordable for whoever wants to consume it (without distorting markets).

While the stated aim of many states in implementing the free mid-day meal in schools is to encourage attendance, there is a more fundamental reason to it. It is in the country’s interest to ensure good health and nutrition of children, in order to enhance their possible contribution to the economy when they grow up (studies have shown that malnutrition and poor health leads to lower educational attainment which leads to lower capacity to contribute to the economy). In this light, it is in the country’s direct interests that children are well fed, and the school is a location where children gather and can be fed (this is where the economies of scale bit comes in). That it encourages attendance is only a positive externality.

Now that it has been established that it is in the country’s interest for the child to be well fed, and that the midday meal in school is a good opportunity to thus feed the child, this presents a classic case for giving a “direct subsidy” rather than a cash transfer. That meals served within school premises are not tradable goods and hence won’t distort markets is a bonus.

Difficulty of Indian Education Boards

With the IITs now having a requirement that students should have scored in the top 20 percentile of their respective boards in order to qualify for admission, we have a chance to evaluate the relative difficulty of various Indian boards.

The IIT Delhi website has the cutoffs for each board. These cutoffs represent the “80th percentile scores” for each board, i.e. if you were to  rank all students who took that particular board exam, these are the marks scored by students 80% from bottom. If you have written any of these board exams and got more than the corresponding 80%ile score for your board, you are eligible to join IIT (provided you score sufficiently in the JEE-main and JEE-advanced, of course).

This plot shows the cutoffs (80th %ile score) for various boards:

Source: http://jee.iitd.ac.in/percentile2013.pdf
Source: http://jee.iitd.ac.in/percentile2013.pdf

Note that the four southern states are on top. These states are reputed to have high educational attainment. Could this be a consequence of easier board exams in these states? We don’t know.

Also, interestingly, these four states are followed by ISC and CBSE, before other state boards. Interestingly, the cutoff for ISC is higher than that for CBSE, which flies against conventional wisdom that CBSE is “easier” than ISC.

Also, if you look at the data, some states have more than one board, and the JEE council has used separate cutoffs for each of these boards. For purpose of my analysis I’ve arbitrarily chosen one board for each state – typically the one whose total is the “roundest” number.