Question

Manish was on his way to an interview. On the way, he encountered his long lost cousin, Vijay, whom he hadn’t met in more than a decade. They started catching up on lost time. Manish learned that Vijay had 3 sons. When he asked about their ages, Vijay replied, “You’re going for an interview, right? Consider this a trial question. Figure out their ages from this: The product of the ages of my three sons is 36.” To this, Manish grumbled that he needed more information. Vijay, then, pointed to a sign board across the street that displayed the address of the area and said that the sum of the ages of his three children was equal to the last two digits of the pin code (zip code) of that area. Manish demanded still more information. Finally, Vijay said, “My eldest son wore a black shirt today. This is all I can tell you.”

What were the ages of the three children?

Solution

Say the ages are a, b and c. We know from clue 1 that a.b.c = 36, where “.” represents multiplication. First step in identifying such factors is to factorize 36 into its smallest factors – 1.2.2.3.3. The next step is to figure out how to group these 5 numbers into 3 groups. Note that we can have more 1s in the factorization. For example, what if 2 kids are 1 year old and 1 “kid” is 36 years old? So, to allow that to happen, we need another 1 in the factors. So the factors are 1.1.2.2.3.3. By trial and error, we recognize that the way to make 3 groups from 6 objects is by making groups of:

1 + 1 + 4 (call it Grouping Style 1, or GS1)

1 + 2 + 3 (call it Grouping Style 2, or GS2)

and 2 + 2 + 2 (call it Grouping Style 3, or GS3)

GS1 requires selecting 1 out of 6 for the first component, 1 out of remaining 5 for the second component and the remaining 4 automatically go into the third component. There are 6 ways to choose the first component, 5 ways to choose the second component and 1 way to chose the third component. This gives us a total of 6.5=30 combinations for GS1. Many of these will turn out to be identical. After some work, we can whittle down the GS1 groupings to the following:

1,1,36

1,2,18

1,3,12

2,2,9

2,3,6

3,3,4

GS2 requires selecting 1 out of 6 for the first component, 2 out of 5 for the second component and the remaining 3 automatically go into the third component. There are 6 ways to choose the first component, 5C2 = 10 ways to choose the second component, and 1 way to choose the third component. This gives us a total of 6.10=60 combinations for GS2. In reality it is easier to work it out by trial and error. After some work, and after recognizing and ignoring the combinations that we have already seen under GS1, we can whittle down the GS2 groupings to the following:

1,4,9

1,6,6

Similar analysis for GS3 gives us no new combinations.

Before we can use clue 2, we need to add up the ages in these combinations. Doing so, we get the following:

1+1+36=38

1+2+18=21

1+3+12=16

2+2+9=13

2+3+6=11

3+3+4=10

1+4+9=14

1+6+6=13

Notice that all the combinations give us unique totals, except for two combinations which both give us 13. The fact that the second clue did not suffice to answer the question indicates that the total of the ages must have been 13. The children could be aged (2,2 and 9) or (1,6 and 6). Any other value for the total age and the answer would have been clear after clue 2.

Clue 3 tells us of the existence of an eldest child. The color of the shirt is immaterial. In the (1,6 and 6) combination, there is no eldest child. There are 2 elder children, who are twins. In the (2,2 and 9) combination, there is an eldest child. Hence, that is the answer. The children are aged 2 years, 2 years and 9 years.

Intersecting Ramanujan’s trail

With the specific problem out of the way, let us think about a generalization to the problem. India’s best known mathematician, Srinivasa Ramanujan, hiked (given his genius, he probably breezed) along a mathematical thought process, probably in 1913, leading to his work on Integer Partitions. In attempting to solve this problem, I seem to have unknowingly stepped onto this trail briefly. Let me explain. Remember that we had to figure out how many ways could 6 objects be grouped into 3 groups. I listed these out as groupings with (1,1,4), (1,2,3) and (2,2,2) objects. There are 3 grouping styles possible, no more, no less. But as the number of total objects grows larger, or the number of groups to create changes, the number of such grouping styles are harder to figure out. At least they seem to follow no simple pattern. For example, to group 6 objects into 2 groups, there are also 3 ways – {5,1}, {4,2} and {3,3}. To group 7 objects into 2 groups, there are only 3 groupings – {6,1} and {5,2} and {4,3}. To group 7 objects into 3 groups, however, there are 4 groupings – {1,1,5}, {1,2,4}, {1,3,3}, and {2,2,3}.

The question is, is there a more general formula to figure this (the number of distinct ways to group N objects into G groups) out? And another question is – what is Integer Partitioning and how is that related to this problem?

So before going into the theory, let us try to do look at this problem from different angles, and we may see an opening to solving it. Let us use the following example – find the number of ways of grouping 7 objects into 2 groups. We discovered (through some mental enumerations, I admit) that there are three grouping styles – {6,1}, {5,2}, {4,3} – possible here. But now, let me draw your attention to a simple fact – notice that the sum of the numbers each of the groupings is equal to 7. Hardly a surprise, you say. There were 7 objects to begin with. And regardless of how we group them, the total object count is 7. Big deal! But it is often rewarding to look at problems from a different perspective. We now know that “grouping 7 objects into 2 groups” is essentially the same as “finding 2 positive integers that add up to 7″. How many ways are there to add up 2 positive integers to get a total of 7? 6+1=7 and 5+2=7 and 4+3=7. No more ways to do it. So, the number of ways to group N things into G groups is equal to the number of ways to add G positive integers to make N. Allow me create a short hand notation to identify this count. I will use P(N,G). P for partition, but really you can use any symbol. I think it only makes sense when N>=G. Otherwise P(N,G)=0.

P(N,G) = number of ways to add G positive integers to make N = numbers of ways to divide N objects into G groups

There are a couple of simple properties of P(N,G) which are easy to observe. There is only 1 way to divide N objects into N groups, and that is to assign 1 object per group. Similarly there is 1 way to divide N objects into 1 group, and that is to place all objects into 1 group. Symbolically,

P(N,N) = 1

P(N, 1) = 1

OK, I think we have beaten that one into submission (if not to death).

Note that we have only defined it. We we after a general formula for P(N,G). And we do not have that yet.

Now, let us visualize this slightly differently, meditate upon it a bit, and see if we can uncover some other properties about P(N,G). The following figure shows the objects, and groupings, visually. This style of representation, called, Ferrers Diagrams, uses one dot for one object. And it arranges the dots in each group along a column. There are 3 groupings shown – these are our friends {6,1}, {5,2} and {4, 3}. If we stare at this for some time we realize that the reason there are 2 columns is because we wanted 2 groups. This means there is at least 1 row, the first row, which has 2 dots in each of the groupings. Now, imagine that we remove those two dots from each of the groupings. What are we left with?

The figure below shows the case where the 2 dots in the first row are removed. We are left with 5 dots. But more importantly notice the groupings that are left. {5}, {4,1} and {3,2}. These are some of the ways you can group 5 objects. This time, we do not necessarily group 5 objects into 2 groups – since we have a grouping, {5}, with only 1 group. It may be interesting to see how many ways there are in all to group 5 objects – not necessarily into 1 or 2 groups, but rather into any number of groups.

The following figure shows all the ways to group 5 objects.

1 group – {5}

2 groups – {4,1}, {3,2}

3 groups – {3,1,1}, {2,2,1}

4 groups – {2,1,1,1}

5 groups – {1,1,1,1,1}

Note that the number of ways to group 5 into 1 and 2 groups in this figure exactly matches the 3 groups in the previous figure. That is P(7,2) = P(5,1)+P(5,2)! That is a breakthrough. So maybe there is a way to break down any P(N,G) into smaller and smaller pieces and add up the totals of the smaller pieces. For example, P(N,G) = P(N-G,1)+P(N-G,2)+ … + P(N-G,G). Let’s try this for P(7,2).

P(7,2) = P(5,1) + P(5,2) = 1 + P(3,1) + P(3,2) = 1 + 1 + P(1,1) = 1 + 1 + 1 = 3

Not quite a closed form, but an recursive solution.

Integer Partitioning and a Generating Function for it

By the way, this is a good place to (finally) get into Integer Partitioning. In the figure above we listed out all the ways to group 5 objects. That is the Integer Partitioning of 5. I use P(5) to represent it.

P(5) = P(5,1) + P(5,2) + P(5,3) + P(5,4) + P(5,5)

Though it is possible to use the recursive technique I described above to solve for P(5) term by term, there is a very interesting alternative way to figure it out. It is based on a near mind-bending technique, based on Generating Functions. It is powerful technique, used in many different places. I have not yet been able to pin down if the abstraction involved in using Generating Functions is pure genius or if it simply falls out of the mechanics of mathematics if only you start from the correct perspective. Let me attempt to convey how Generating Functions work in this specific situation.

We want to find out all the possible ways to group N objects. But instead of starting with this specific problem we turn the problem on its head and solve a much more general problem. Let us use a bunch of 1s, a bunch of 2s, a bunch of 3s etc. and see what we can add them up to. We do this exhaustively. That is, we we do not let any combination fall through the cracks. Then, we can count up all the different ways a bunch of addends give us the total N. For example, if N is 5, and we do this exhaustive (but structured) adding of all integers (repetitions allowed) and see how many combinations add to 5. We can be sure that none of the addends will be 6 or greater. Similarly, the number of addends will also be no greater than 5, because each addend is at least 1, and with 5 1s we already are at 5. So, the number of combinations we need to look at only needs to cover the addends 1 through 5, and no more than 5 of those addends.

This exhaustive search seems hard to do. But generating functions provide a structured method to approach this. Let us do it one step at a time. We will look at the potential contributions to the total, of each candidate addend one at a time.  Let us look at the addend 1. What totals can 1 contribute to? 1s can add up to 1 in 1 way, {1}. 1s can add up to 2 in 1 way, {1,1}. 1s can add up to 3 in 1 way {1,1,1} and so on. That is, 1 alone can add up to any N, AND, in only 1 way for each given N. In fact if there is no 1 in the addends, then 1 can contribute 0 to the total. In generating function terms this would be represented as follows:

What? Where did that come from? That is typically, always the reaction in my own mind when I see that. Essentially, the order of , which is k, indicates the total that the addends (remember we are only talking about the addends being 1) add up to. The coefficient of , which is 1, indicates the number of ways the addends can be added up to get to k. There is only 1 way to do this using only 1s. And there may be no 1s in the addends, and that is the term .

But what if 2s are allowed? There can be 1 two, or 2 twos, or 3 twos, etc. Thus the generating function based on contributions only from twos is as follows:

This basically means twos can add up to 0 or 2 or 4 or 6 or 8 etc.

Now, we know that we can have both 1s and 2s in the set of addends. In fact, we can have 1s, and 2s, and 3s, and 4s etc. Let us restrict the addends to 1s and 2s for the time being. There can be 1 one and 1 two, or 1 one and 2 twos or 2 ones and 1 two etc. How do we mathematically express these combinations? This is where the true import of the elegance of generating functions becomes clear. By allowing the contributions of the addends to be in the power of x position, products of and correctly give us the sum in the power’s position, .  Further, if there are multiple ways to add up to a given addend, they show up in the coefficient position. That is, as an example, say we want to figure out the Integer Partitions of any number N, using only the addends 1 and 2, we can multiply the individual addends’ generating polynomials.

This product of infinite polynomials will give us every possible way to add up 1s and 2s to get us to a specific total N. Say N=5, then we know we need not worry about and above in either polynomial.

And after some math, we end up with the necessary coefficient for , and that is the number of ways to get to 5 using only 1s and 2s.

Now, just extending this to allow contributions from the addends 3, 4, 5 etc., we get the full generating function for Integer Partitioning.

P(N) = coefficient of in

Whew! So we have that out of the way. What amazes me no end is how generating functions are able to hijack this polynomial product form to solve (or, more accurately, bring structure to) seemingly impossible scenarios that need counting. Notice that we still do not have a closed form for the Partition of Integers, P(N). Computers can be employed for the multiplication of terms and accumulation of coefficients.

A Closed Form Approximation

In 1918, Srinivasa Ramanujan and his advisor, G. Hardy, came up with a closed form , albeit a closed form for an approximation to P(N). With the current understanding of Integer Partitioning, this seems like an amazing accomplishment. This closed form was:

This produces pretty good approximations. For example, P(100) = 190,569,292, and this closed form approximates it to around 199 million.

References

1. Joseph Laurendi, Partitions of Integers, 2005, http://www.artofproblemsolving.com/Resources/Papers/LaurendiPartitions.pdf

2. Wikipedia, http://en.wikipedia.org/wiki/Partition_%28number_theory%29

“Ammamma…Boost…”, I would ask her for a hot malt beverage, as she would get busy in her tiny kitchen after her short afternoon nap on the hard concrete floor, with a strategically placed pillow for her head. Ammamma means maternal grandmother in my mother tongue, Telugu. Amma is mom, and ammamma is, literally, momom. She would then get busy preparing the late afternoon coffees for the elders, starting with the eldest – Tatagaru, and Boost for the kids. Although I mostly saw her only over summer holidays, this particular aspect of her routine was probably eternal. In fact, all her routines were seemingly eternal yet inexplicably fresh every time. She would hunker down at the old, grime-laden, two-burner gas stove sitting on the floor of her tiny kitchen and with what seemed like an impossibly tiny collection of utensils, groceries and gadgets, came up with the most exquisite of dishes. Simple fare it always was, and she was not a great cook, but the taste of her cooking was earthy and heavenly. Vegetables of all manner were shallow fried. Coffee and Boost was not served before being poured several times, alternating between two tumblers to generate froth (steamed milk). “Boost tagutawa, Kishtappa, aain?”, she would ask. The “aain?” was kind of like Amitabh Bachchan’s pan-laden mouth confirming something – “aain?”.

Kishtappa was her pet name for me. And she had a way with all children. She was our story-teller come in-house-magician come mystery-keeper come stand-up-comedian come play-pal, all rolled into one. Above all, she was our friend. Maybe because she had only very little formal education (“Appatlo, SLC pass aiyyenu”, she would confirm – “I completed 8th grade in those days”.), or maybe because she was married off when only 9 years old, my grandmother never really got along as well with adults as she did with kids, and always seemed to have stamina for reliving the childhood which she did not get to truly enjoy. She had a way with kids. And with five daughters and three sons she never ran out of kids or grandkids or great-grandkids to enthrall.

Ammamma was the queen of her territory in my Tatagaru’s large house in Dondaparthy, in Visakhaptnam. Her territory included her tiny kitchen, the attached utility area (her bathing area), the verandah just outside the kitchen where we would all eat (sitting on the rough concrete floor), a step down area from the verandah where the dishes would be washed and which led to the back yard, the strip connecting these rooms in the back of the house to the front verandah of the house (at the head of this strip was where she would strategically place her pillow ever afternoon for a watchful nap), and, the sacred sanctum of her territory, the store room (ammamma kottu) which remained mostly under lock and key. Except at some strange hours at the night. This was truly ammamma’s territory. I have probably only caught a glimpse of the inside of this room a handful of times, and I think I have been lucky enough to step into the room only once or twice. The room was just a large walk-in pantry with wooden shelves loaded with boxes and bags of all sizes and descriptions. The occasional mice, cockroach, or a scorpion (none of which would faze the dauntless ammamma) along with the dim lighting in the room and ammamma’s reluctance to let kids enter added to the eeriness of that store room. On the rare occasion we kids found the room open and would gather enough courage to peek in, ammamma would bustle out form the dark confines of the room, shoo us kids away, and tell us kids to stay away from the ghoul-laden innards of the place. She would then promptly hand us 25 paise or 50 paise and ask us to go get ourselves some mint candy (straangu billalu) from the Nair kottu – Mr. Nair’s small shop which sold everything from candy to biscuits to soft-drinks (of which, goli-soda was a strange attraction) to cigarettes to stationery. It was a shop where it seemed you could find anything that could be held in one hand.

Ammamma had a very interesting relationship with her husband, my Tatagaru (respected grandfather). To us she seemed to never be in speaking terms with her husband. I have never seen her converse with Tatagaru. However, her day would revolve around him. Being a lawyer, Tatagaru would spent some hours of the day at the Vizag District Court. And the rest of the day, his clients would come see him at his elaborate home office. He was a simple man, often sporting only a dhoti (similar to Gandhiji), but his office was well endowed, with lots of Law books and legal proceedings, case studies etc. He had an office room, which was separated from the road outside by a small waiting room for clients to wait in if he had company. Ammamma would have to keep an eye on Tatagaru, for he was a man of few words and if he needed lunch or one of the many servings of coffee, she would prepare that. She also had to occasionally keep an eye out for important clients and make sure they were served coffee or water etc. She was particularly displeased with lady clients who would spend a lot of time in Tatagaru’s office and would keep an eye on them as well. The office room was always open and though there was nothing to be concerned about, I guess she was possessive after all. These lady clients were often from small villages, and were from farming families. They would be visiting Tatagaru in relation to some land-related case. Often they would wear only a saree without a blouse to go with it, and this seems to particularly displease ammamma, although she would never make a big fuss about it. At most it would be an extra visit to the office to remind Tatagaru, “Vadinchestaanu, kallu kadukkoni randi” (“I will serve” – likely lunch – “wash your feet and come”).

I felt that the best times ammamma had was with her grand kids. With her kids, she was probably too young herself. With her great-grand kids, she was too old. But the best times she had was with us grand kids. She would play and teach us card games and tricks, then she would tell us stories or read us stories from the children’s magazine “Chandamama”, or read us jokes from her Telugu magazines, such as “Swathi”, “Andhra Jyoti” or “Andhra Prabha”. She would play “ashta-chamma” (similar to Ludo) or some variant of knucklebones with us kids, and perpetually keep us enthralled.

Maybe it was childish innocence. More likely it was sagely wisdom. Ammamma was never one to get too emotionally attached or emotionally charged. Her actions were devoid of any scheming, but like I said, probably not out of innocence, but rather out of a sagely understanding of the futility of such schemes. She was happy-go-lucky. She might have been different when younger, but as she grew older, she became a person of caricaturable simplicity. Maybe that is why we kids loved her. We could understand her. She would get happy at the small things in life, while the elders were never truly happy with anything. Some small things I remember about her were the way she would comb her tiny swathe of hair very meticulously each day after her bath and after applying a generous portion of hair oil. She would neatly part her hair at the center and with a vigor exceeding her age she would straighten her hair on each side, before finally tying it up into a braid or a knot. That she managed to do all this with a tiny mirror, and a tiny comb, each less than a few inches in size, never ceased to amaze. She loved the cinema, and would take us kids and go watch movies, mostly mythological or allegoric ones. She took me to watch “Keelu Gurramu” (the Magic Horse) once. For someone who enjoyed mythological movies, she was not much into devout worship or religious rituals. She lived an uncomplicated, pragmatic life. Although some of her beliefs would be considered backward (such as resisting graduate education for her girls), her practical point of view was that it would get harder to find a good match if the girl were overly educated. Regardless, and thanks to the calming influence of Tatagaru, most of her sons and daughters got a good education, some going on to complete their Masters, and all others picking up Bachelors’ degrees.

Ammama visited Bhilai, the steel city that was my birth place, and where my father used to work, at least a couple of times. And for her, visiting Bhilai was a wonderful time off. A vacation. Once she had a cataract operation (I think it was her left eye) at Durg (a town near Bhilai) and she was very pleased with the outcome. The other eye, which was operated upon in Visakapatnam itself was never quite the same, she said. Regardless, ever since her eye operations we only remember seeing her with thick lensed spectacles, with a thick black frame. The only time we’d then see her without her glasses was during her afternoon naps, and even still, the glasses would be tucked neatly under her pillow, lest we kids stepped on her only pair while running around during our afternoon games.

Ammamma was born Adibhatla Venkata Ratnamma on May 5th, 1921, in the hamlet called Dimili Agraharam, near Elamanchili town in present day Andhra Pradesh. Her father was Sri Adibhatla Suryanarayana, who was an inspector in the Revenue Department, and her mother was Adibhatla Perindevamma. She had two elder sisters and an elder brother. When she was born she was grossly underweight (likely, very premature) and her mother had given up hope about her surviving. The newborn did not have enough strength to even suckle milk be it from the breast or bottle. The elder sisters would go and get milk from other nursing mothers in the village and feed the newborn using a cotton wick. This was the only way the sickly child would ingest food. After three months of such feeding the mother was finally convinced that ammamma would survive. She survived. And how.

The elder sisters died in their early twenties, probably during childbirth. Ammamma, though born sickly and underweight, outlived all her other siblings, and she led a blessed, healthy life. She was married to Sri Tata Sri Rama Murthy, my Tatagaru, on March 16th, 1930 in “Kanukurthi vari satram” in Vijayanagaram city. It was a 5-day long wedding, complete with city-tours aboard a pearl-encrusted palanquin. Henceforth she became Tata Venkata Ratnamma. Married to someone destined to be a renowned lawyer, blessed with loving children and grandchildren, living a exceptionally healthy life, managing to keep her distance from petty attachments while retaining the ability to stay happy, ammamma had a great life. Recently, she had her first great-great-grandchild. Her second daughter’s first daughter’s daughter had a daughter. Truly rare. If we go looking for any misfortunes in her life, the untimely loss of her eldest son-in-law is probably the only one.

Ammamma passed away about 4 hours ago, sometime between 5:15AM and 5:30 AM on September 2nd, 2010 (India time). She was admitted to the hospital in the early hours of September 1st when she seemed to have lost consciousness after a few days of minimized food intake. Her blood pressure was quite low. In the hospital, she regained consciousness, recognized people, recognized the doctors and asked about all her grandkids. The IV drip helped her improve her blood pressure to near normal levels, and the oxygen mask helped her weakened heart. She seemed to be on her way to a steady recovery. Finally, it was her kidneys that failed. A woman who was so strong in her life, by her 90th year had gotten really good at putting up a stern fight against and evading any illness. She was a throat cancer survivor. And just as she gave us all hope that this would just be another of her minor illnesses, which she would fight down handily, she pulled off her last magic trick. She decided to say goodbye without any drama, without any emotion, without any inconvenience to others. Kavita and I must have been at the ISKCON temple in Hillsborough, North Carolina chanting “Hare Krishna, Hare Krishna, Krishna Krishna, Hare Hare … Hare Rama, Hare Rama, Rama Rama, Hare Hare” right when, for the last time, Ammamma must have said her favorite prayer one last time – “Krishna Vasudeva”. On our way back, we got the call.

Ammamma left us on Krishnashtami (Lord Krishna’s birthday), and left us with happy memories.  There is not a single negative memory I have of her. That is probably how any grandson feels about his grandmother, and she was the only grandmother I knew. But still, I feel that she was special. She was different. Knowing her was a boon for me. And like Kavita reminded me, Ammamma was the only grandmother she knew as well. Kavita shares the same kind of happy memories with Maamma (as she called Ammamma) as I did. Ammamma taught me more than she will ever know. Or, maybe in her own way, she did know all along. Happy-go-lucky. Happy-went-lucky.

Acknowledgements: Some of the memories and Ammamma’s biography are based on recollections by my cousin, Prasad. Most of the pictures are from Anant.

Dr. K. Ramamurthy (my dad’s uncle) wrote the following in response to this issue.

This gave rise to some thoughts which� I present here.

I agree that educated incompetence is a real phenomenon. I see it at work, at school, and even within our family. Typically, it manifests itself as a minor irritant with small repercussions; however, on occasion, it can bloat into a dangerous phenomenon, affecting life-changing decisions we make.

The people who think that they know (or, more accurately, who have been conditioned to think they ought to know) feel this pressure to pretend that they know. They may not know, but their ego does not let them accept or project their ignorance.

Now, take the case of a person who is formally uneducated. That person may not have any degrees, but may have sufficient common sense and, indeed, an abundant supply of humility. Unfortunately, lack of formal education, brings lack of confidence in addition to the abundance of humility. Abundance of formal education brings abundance of confidence (over-confidence, in most cases) and a lack of humility. The best kind of education encourages the right amount of confidence, and the right amount of humility.

That begs the question, “What is the fundamental problem with formal education?”.

Formal educational arrangements often cannot give one-on-one attention that is necessary to truly explore the limits of an individual’s potential. The result of this fundamental clash between individual potentials and group educational-arrangements is a lowering of the education bar. Further, in order to smoothly impart knowledge to a group, and then to smoothly measure the efficiency of that process, a common canonical framework (the box) must be constructed. All formal education builds within the confines of this framework. This explains why, in exams, typically, all questions have a reasonably fixed answer. Mathematical problems, typically, only provide precisely the data necessary to solve the problem. No unnecessary data is provided. In Chemistry lab, only the reagents and equipment necessary to prove the workings of a certain reaction are provided. To me it is not at all surprising that relying only on formal education as a means to educating yourself is bound to teach you a canonical view of the world and its problems. It teaches you to construct the boundaries of your sandbox of thoughts before trying to solve a problem. It teaches you that when you have a problem you just need to use the resources readily visible within this sandbox. It does not, typically, teach you where to look for resources to solve a problem. It does not teach you how to get rid of the sandbox. And most significantly it does not teach you how to look *for* problems. Problems are assumed to be handed to you, with your job being restricted to looking for solutions. This is a fundamental problem with all forms of formal education. And, if you think about it, this formal educational structure is a direct result of the simplifications that *must* be made for any, large-scale, practical, hand-off of human understanding from one generation to the next. That is, I am not blaming the formal education. I am saying it is doing exactly what it was meant to do. To identify and solve real-life problems, it helps to know how to solve a canonical, artificially-created, abstract versions of the real-life problem. Formal education signed up to distill real-life problems into canonical problems and teach us how to proceed from that point further. It did not sign up for more.

Now, the responsibility of the second half of education – the ability to identify a real-life problem (before it is handed over to you on a platter), the ability to whittle down a real-life problem into a canonical form, the ability to identify parameters that affect the solution to this problem from a much larger selection than you are used to dealing with – lies with the individual. Even if we could, in a perfect society, provide one-on-one attention to each student, human brains are not very good at communication of complex thought. The only brain that can efficiently sift through the deluge of complex thoughts that arise inside it, it the very same brain where these thoughts arise. This ability to learn by looking inwards has been called meditation, self-awareness, reflection, wisdom, and, common sense. This is an ability that cannot be taught by formal educational tools. It is, therefore, a resource available to the educated and the uneducated in equal measure. The uneducated person, probably enjoys it in purer, undiluted proportions. The formally educated person may, unfortunately, end up suspicious of such free-ranging thoughts arising in his or her mind and proceed to quell them. This may explain why educated people often appear to be unimaginative and conforming – thinking “inside-the-box” out of habit.

Maybe this makes a case for a statutory warning to be legally enforced on all formal educational fora – FORMAL EDUCATION, HOWEVER ADVANCED, IS ONLY RESPONSIBLE FOR A SMALL FRACTION OF YOUR OVERALL EDUCATION

(PS: By the way, be careful not to interpret this to mean that everything your maid-servant says is wise.)

I read some more interesting insights form Dr. Ramamurthy a few emails down the chain.

And here is my response:

In fact, rational thinking is a necessary condition to allow out of the box thinking. It is not a sufficient condition though. A bit of inspired (some may even call it non-rational, creative or artistic) thinking becomes necessary as well. This is that spark of subconscious creativity that conscious rational thinking can then give form to.

All this talk of rational vs creative, boxed-in vs out-of-the box thinking reminds me of the following incident which happened recently.

"No, K. The problem is not that I use only my left brain. The problem is that I can only use the brain that is left."

The question I posed to myself after thinking through the puzzle was, “How can we divide a square into 7 equal parts with only a straight edge and a compass available?” Note that the question implies that we do not have a ruler or a scale. We have a straight edge, but without any markings on it to indicate inches or centimeters. Even if it did have the markings, such markings can only measure accurately up to a certain level. For example, say you have a scale with markings at the granularity of a millimeter. Say the square had a side equal to some irrational number, say or , or, even a simple integer which is not a multiple of 7, such as, 8 millimeters. There is no way to measure or or millimeters using such a scale.

There are at least 2 approaches to dividing this square into 7 parts. The first is a simpler approach and the second is slightly more involved. Let me talk about the second one first. The first one will then become easy to see. The main intuition behind the first idea is that a triangle’s area depends only on its base and height. If we can mark out 7 equidistant points along the square’s border, thus creating 7 equal bases, we can join the bases to the center of the square to create 7 regions with equal areas. The heights of these shapes will be equal, and the bases are equal by construction.

As shown in the figure to the far left, the corners of the square are joined, and the intersection gives us the center of the square. Then, assuming we already have the 7 equidistant points of the square’s edge marked out,  joining the center to the 7 points creates 7 regions with equal area, as shown in the figure to the immediate left. It may not be obvious why these regions are equal in area. If you divide up the quadrilateral regions (regions with 4 edges) into 2 triangles, you should start seeing that the area of the quadrilateral piece is the same as the area of a triangular piece since they both have the same height (, where is the length of the square’s side) and same base ().

The question then is, how do we make those 7 markings on the square? Let us first imagine that we peel off the edge and lay it out along a straight line. If there were some simple way to then divide the peeled out, straight line edge into 7 equal parts, we can visualize rolling the laid out edge back on to the square. Of course, the actual process would involve measuring the widths of each piece using the compass and then laying them back on the square’s border. At the corners these segments may have to be further broken down into two pieces. How to do that it is relatively intuitive with a compass and I will not go into explaining that. The next two figures illustrate the process of peeling off the edges of the square and laying them onto a straight line, and the process of rolling the laid out edge back on to the square once the 7 equidistant markings have been made.

The question now remains, how do we divide the laid out border into 7 equal pieces. More generally, this question can be posed as, how can we divide any line segment into n equal segments.

The figure to the left illustrates the steps. The steps are labeled (a) through (e). In Step (a), we draw a line using the straight edge at some angle to the line segment to be divided. This edge is shown in a light color and its length does not really matter. Then in Step (b) make 7 equidistant markings on the newly drawn line segment using the compass. The length between the markings does not matter. Just pick something that seems to fit in the space. Then, in Step (c), the 7th such marking and the end of the original line segment are joined. In Step (d), using the compass, identify a point corresponding to the 7th marking, on the other side of the base line segment. Connect the ends of the base line segment to that point and complete the paralellogram shown in the figure in Step (d). Using the compass mark out the other 6 markings on the line on the other side of the base line segment. This is shown in Step (e). Join the corresponding intermediate markings using the straight edge. Voila! The line segment we wanted to divide into 7 pieces just got divided into 7 equal pieces.

Now, by retracing the steps laid out in the figures above, you can divide the square into 7 equal regions. This question is also posed as, “How can you divide a square cake into n equal sized pieces?”. Now that we know how to divide a line into n equal parts using a straight edge and a compass, we may realize that there is a simpler way to divide up the square cake. Instead of dividing up the whole border into 7 equal-length line segments, why not just divide up one side into 7 equal length segments. Then we can cut the square into 7 long rectangles, in a fashion similar to the one proposed in the original puzzle (see the pdf slides posted at the beginning of this article).

This also brings up another interesting question, although quite off-topic, it may seem. We have been using the phrase straight edge throughout this article. But what is straight? How was the first straight edge made? My guess is that a taut string might have been used to define straightness. But then what is space itself is curved? Does that mean a taut string is also actually curved in accordance with the space it is in? What does it even mean to say that space is curved? Curved with respect to what? What is the underlying measure of straightness, against which the space appears curved? There are probably questions for another disussion. I do not know the answer. I will have to think and read a bunch before it may be clear. It may, of course, never be clear. There is no reason everything should make sense. Nature is not obligated to be understandable by humans, much less one particular human that I call me.

“During my management consultancy days, I’d start my classes with “Three Ds”: Discipline, Dedication, and Devotion.

We have to start anything in life, commencing with our earliest education, with the rigor of Discipline: regulated studies in terms of time allocation, understanding of what we study, practicing to become perfect, and humility not to be carried away by early successes (or depressed with early failures). You keep at it in spite of obstacles to reach the goal you set for yourself.

When you’re sufficiently integrated in a disciplined web of working (doing things), you get to the stage of Dedication – a stage where you totally, intrinsically, get merged in what you do and what you want to achieve. You breathe, live, and think all the while of your chosen field and its nuances to be able to excel. That’s how great writers, poets, scientists, musicians, innovators, nation builders, and freedom fighters like Gandhi dedicated their whole life to a chosen cause.

From this stage comes Devotion, where your “life and work” become a Religion unto itself. That’s how Thyagaraja, Purandara, Meerabai, Aurobindo, CV Raman in our life time and people like Einstein worshiped what they chose to do. All the greatest achievers have gone through these stages, knowingly or unknowingly.

Consider the training of today’s top notch players who reach the very top, their journey begins at very early age and goes on unhindered and unfettered for several years to reach the top. Certain failures are inevitable during this long journey but they’ve to trod on incessantly to reach the peak.

Of all who tried, the number who did or did not make the final assault is immaterial. The very process and trial is ennobling – in fact, religious. It’s like seeking the elusive God, but there is bliss!!

In such pursuit, the teacher becomes the most important being in our life. It’s said in our scriptures that one cannot attain the highest pinnacle without a “Teacher.” In our daily prayers, we do give homage to our teacher: Guru Brahma, Gurur Vishnuhu, Gurudevo Maheswaraha, Guru Sakshsat Parabrahma, Tasmai Sri Gurave Namaha! Discipline starts with respect to the teacher – starting from our parents who are our first teachers, to others who have taught us, guided us, helped us, sustained us, given solace in our trials and difficult patches, and remained our “guiding lights” throughout our life.

Unfortunately, the teacher-taught, trainer-trainee, professor-student, employer-employee relationship has become now too commercialized to nurture a meaningful, respectful, disciplined way in life. Without this kind of moral and ethical approach, the society declines. It’s only the few chosen (by whom, I can’t say!), who are able to fuse the 3-Ds to be the Great in their individual life!!

By our performance, we’re not ONE of those.”

His observations on how to simultaneously achieve happiness (selfish motive) while at the same time being productive to the society (altruistic outcome) by following the course of discipline, dedication and devotion speaks to me and I hope to many of us. He elaborated in a later email thus.

“To further elaborate, the first D is the base or foundation on which the second D, dedication, is superimposed. The third D, Devotion, is necessary, along with the other two, for the final outcome, or assault, as it were. That is reaching out to the pinnacle. While the first and second Ds have a continuous nature, the third D could be even ‘momentary or fleeting’ but it’s that fleeting moment – like in deep, prayerful, thought that gives the final ‘push’ and the ‘answer.’

This is referred to in our books of lore about ‘Rishis’ in deep meditation; we see this in our scientists and researchers in their hour of ‘discovery.’ Philosophers of lore were of that genre.

Recently I read of an interview of Dr. Ramachandran, the Neurosurgeon-researcher and author of books on brain structure and functioning. He was alluding to his conversation with Chembe Vaidyanatha Iyer, doyen of Carnatic Music and said that while the music maestro was rendering a raaga and aalaapana, he was ethereal, as if he was in ‘devotional ecstasy.’ At that moment the maestro was not aware of his surroundings, the visitor, or anything else but his music rendition. That is the moment of the third D.“

Change of Plans

Initially, we had planned that only Kavita would go to India, and I would just meet her in London on her way back. However, getting a Visa for UK was a mess. We were required to go to New York for fingerprinting, according to some new rule started in April 2008. Also, we’d have to figure out exactly where we would stay in London etc. even before applying for the Visa. Ludicrous enough as it was, it was made worse by the fact that travel to New York was quite expensive too. We dropped the London plan and decided to fly non-stop from the US to India, bypassing Europe altogether. This would avoid all transit Visa problems – many cities in Europe require transit visas even if you are not leaving the airport.

Preparation

Kavita handled the entire process of searching for fares and dates, reading out our XXL (Extra Extra Long) names to each travel agent going “M as in Mary, U as in Umbrella … ” and so on, and finally settling for the best deal she could manage. The journey was quadruped – 4 legs – Raleigh to New York by Delta Airlines, New York to Mumbai by Delta Airlines’ youngest aircraft a Boeing 777 200LR, Mumbai to Hyderabad by Vijay Mallya’s Kingfisher Airlines and Hyderabad to Visakhapatnam by Godavari Express. The process of packing was broken into two phases. The first, drawn out, phase was to figure out what to carry and then dump it in the study room. This included small gift items we had been collecting over some time, our cameras, old clothes we wished to give away in India etc. The second phase, consisting of cramming all this into our suitcases, weighing them and rebalancing them for weight restrictions, took only a few hours. During this process, I spent some time to calculate how much money it costs (not to us, but in terms of the fuel burnt but the airplane) to carry each extra pound and came up with something like 40 cents given the distance we were traveling and the price of jet fuel nowadays. Although this seems like an irrelevant calculation, especially given that we were carrying only 75% of what we could carry without paying extra, I used this piece of information to convince Kavita that we should not carry a relatively heavy utensil (kadhai) which she wanted to take to India and give to somebody there.

Kavita + Vacation = Excitement

Kavita says that she is destined to never be able to travel without incident, and this time the incident that started the excitement was someone stealing the oxygen sensor from Kavita’s car just days before we were to leave. We managed to drop it off in the repair shop before we left, and used my car for all travel needs, including transporting our potted plants to a friend’s place. Then everything seemed to go on smoothly until the day of travel. We were all set on June 29th evening. Balan, our friend, dropped us off at the airport, we got our boarding cards, breezed through security and found ourselves sitting in the plane. Then, after about a half hour in the plane came the news that the weather in New York would not allow our tiny craft to land, and so we would not bother taking off. The high price of fuel had finally caught up with our plans! Fuel prices forced Delta to switch to a smaller craft, and a smaller craft could not handle the weather like a larger one could have; hence, we had to right shift our entire travel plans, including the last leg of the journey from Hyderabad to Visakhapatnam by train, by one day.

We decided to not go back home with all that luggage. Instead we picked to a motel near the airport and decided to spend the night there. The next day we had a relatively early flight to catch. The Delta agent, upon our flight’s cancellation helped us with our tickets for the next day. She was on the phone for almost an hour trying to rebook us. Getting us rebooked on the Kingfisher Airline flight from Mumbai to Hyderabadwas what took the agent the longest, as she was finding it hard to get in touch with them. Eventually, we we put on the flight form Raleigh to Atlanta, and then we were to catch the international flight at its origination point, Atlanta, rather than chancing another flight cancellation the next day. Meanwhile, the lady also informed us that our Kingfisher flight between Hyderabad and Mumbai, during the return journey, had been cancelled for some unknown reason. She booked us on a different flight, one day sooner than we wanted to leave from Hyderabad. Thus with our vacation kind of squeezed in on both its ends, we settled in the motel room, hoping this would be all the excitement we would be served up. What did we know.

We dined at a Waffle House near the motel. The omelet was a bit greasy, but the toast and hash browns were great. Back in the hotel, we again resorted to technology, to fix the mess caused by fuel. Indian Railways now has a half-decent website where you can cancel and book train tickets. The website is non-intuitive and crashes often. However, it is a remarkable improvement compared to how things were just a few years ago. I was able to cancel our existing reservation and get a reservation for the next day. It was a different train, however – Visakha Express instead of Godavari Express, and I had to reserve “Tatkal”, which is kind of like last minute reservations, which cost more. We should consider ourselves lucky that were able to get reservations at all, in the AC (air-conditioned) 2nd class compartment. Trains in India have different classes of service -AC 1st Class, 1st Class, AC 2nd Class (also known as AC two-tier), AC 3-tier Sleeper, 3-tier Sleeper and General (unreserved comparment). Compared to the US, fares are much cheaper. With the amount of luggage we had, we wanted to at least go with AC 2nd Class.

India

The craft that flew us from New York to Mumbai was Boeing 777 200LR, the youngest craft in the fleet operated by Delta. The seating was noticeably more comfortable than my previous experiences. There were fewer seats across the width of the craft, an arrangement of 3-3-3 instead of 3-4-3, giving everyone a few extra inches. Upon reclining, the backrest moved back as usual, but the base of the seat, which typically is stationary, also moved forward a bit, thus making it more comfortable. The in-flight movie selection was extensive. I saw many documentaries, and I especially recall one called “The 11th Hour” which was about global warming and how we are close to the tipping point of a headlong dive to no-return.

Mumbai’s Chattrapati Shivaji airport was under renovation. After customs and getting our bags, we took an internal airport bus to the domestic terminal. We had an overnight wait there, with our flight to Hyderabad scheduled for early next morning. Kavita had made many friends along the way, starting from Raleigh, and continuing making acquaintences on the flight. They all sat together in the waiting area, while I tried to find a quieter corner to try and catch some sleep. I realized that I am getting better with my ability to slow down my heart rate and rest, even while seated. I used to be a chronic failure when it came to sleeping while seated. Anyway, the few minutes of sleep I could catch here and there were enough for me to get back into the timezone. Next morning, we started with a machine supplied NesTea from a vendor at the terminal. The tea was not anywhere as good as I remember tea in India being, maybe because it came from a machine, but more likely because they used creamer instead of real milk and mixed up the proportions. I was also shocked by how the cup sizes for beverages had shrunk. Either that or I have gotten used to the mega-mugs of the US. But later in the trip, in a Hyderabad bus stand, where I would finally find some better tasting, authentically prepared, non-mechanized, tea, I would be shocked by the audacity of the shrinkage. I would find the size of the cheap, plastic, tea cup was exactly the same as the size of the cup you get with Benadryl or Robotussin cough syrup. It could probably hold no more than 4 tablespoons of tea. Maybe, that is way it was priced at 4 rupees.

Anyway, we made it to Hyderabad that morning, where my friend CK picked us up. After freshening up and being deposited into the Visakha Express train, we reached Vizag next day and thus began the actual vacation. It was the year we went to Bhilai – Kavita’s second trip, her first was when she was in the 5th grade.

(Sorry for an abrupt ending to this article. I started it in earnest right after this trip, but did not finish it and forgot most of the details of what happened. Just putting it up so this writing, even if partial, is not lost.)

Living in a tiny square patch of a town, in the middle of thousands of square miles of featureless flatlands, encouragingly named “Normal”, was expected to be hard for this single woman. She took to it like a duck to water. Having been on the phone with her almost every one of the days in these 5 years, I wouldn’t blame you if you thought that we were born with the cellphone on our ears, instead of the proverbial golden spoon in our mouths. I saw her develop lasting friendships. I saw her live an independent and brave life. I saw her work hard and cheerfully. I saw her build her life in her tiny, but beautiful apartment in Cardinal Court. Her friends, her teachers, her attitude and her spirit are all worthy of our thanks. She has done us all proud. She has given us hope and promise. Dedicated work, making the best of the opportunities you get, will get us rewards. Her mother, my mother-in-law, is to be especially congratulated. Her words, which usually carry a ring of innocence, humility and simplicity are in fact words of great wisdom. Her role in letting Kavita be herself, more than a direct hand in guiding her educational growth, is one important reason for Kavita’s achievement.

My role has been one of an honest and eager listener. Although I find very few understandable English words in her presentations or reports on her subject, I always tried to atleast keep myself aware of the basics of her field. Other than playing the joker to relieve both our stresses at the end of the day, my role has also been to positively appreciate the praise-worthy in her efforts, and critique the fallacies. I have always respected her work, her efficiency at it in her fruit-fly infested lab, her action-oriented approach towards a plan and the celerity with which she picked up something new. I have been always motivated by the positives in her. And this is a moment to reflect on these 5 years and to look towards the future as she puts the knowledge she has earned to good use. Congratulations, Dr. Vadali. You deserve it.

Illinois is a place I visit often. Kavita and I look forward to meeting up on long weekends because they are a whole day longer than a regular weekend. The plan is always to meet up and spend time with each other. The way it ends up is we spend time with friends or family. And fortunately, both of us enjoy and work best in that mode.

For this September, the plans had been drawn over a long period of time – several hours spread over several weeks. That’s a big part of the fun; planning what we would do, once together. The plans, like I said earlier, always include meeting with friends and spending time with them, other than going some place nice and of course, some small doses of shopping.

The plans for this Labor Day, however, included an unusually enthusiastic attempt at meeting with many of my undergraduate schoolmates who studied with me at IIT Guwahati and now lived in or close to Illinois. The surprise was that I met up with all those friends I planned to meet, and more. Kavita, who enjoys being around people, loved every minute of it. And I loved it all the more. I was thrilled to see these wonderful people, and seeing Kavita’s delight doubled the effect.

As I write this, the American Eagle flight from Bloomington, Illinois, to Chicago is accelerating on the runway to gracefully dive into the late afternoon haze. A pale early fall sun sweeps over the corn fields that stretch as far as the land is visible, which is no too far since very soon the misty haze takes over the flat geography and stretches farther and upwards and leaves you looking into the deep blue of high skies.

The flight’s well on its way now. The vast flatlands of the American Prairies are my first memories of this country. 1999, August, when I landed in Indianapolis, Indiana, I was taken aback by the blueness of the skies, at the stillness of the air, the range of your eyes can scale with a glance across these lands and the emptiness that stretches between scanty pockets of civilization. And that welcome feeling always comes back when I return.

This time the vacation was more fun than usual and that is a good thing. The downside to that, inescapably tied to the by-product of Einstein’s laws of relativity, is that the more fun you are having, the faster time flies.

I started from Raleigh on a Thursday evening, straight form work. My accomplice in this successful escape was my trusted friend and colleague at work, Srini. Of course, my ever- supportive manager, Ken, knew that I would be taking off a bit early on Thursday. In fact, he took the whole team to lunch to P. F. Chang’s, Chinese Bistro, the same day and he did say, “You guys have been working hard. You should take off after this lunch.” He did not realize that I would take the “take off” part literally.

Kavita walked in through the main entrance of Bloomington airport, just as I approached the same place to pick my baggage up. This did strike me as a reasonable coincidence. One of the many I would witness in this trip.

On a few occasions in my life I have overeaten consciously, and enjoyed every bit of it. Once was when in Vizag, Lakshmi akka (sister) had made Inguva chaaru (Rasam with Hing). One was when Kavita made phulkas that evening for dinner. These are just rotis made with whole wheat flour; but the trick is that you put it directly onto the gas burner’s flame after baking it for a few minutes. It puffs up into a ball in a few seconds and it is ready. Kavita had learnt this culinary skill recently from Venkat’s mother, and I can vouch that the teacher and student have done well.

I had not taken Friday off and neither had Kavita. In fact I had a good bit of work to finish. I used Kavita’s laptop to connect to work. After tea and breakfast in the morning, I dropped Kavita off at her lab, came back to her apartment and started working. Lunch was at Coffee House with Carie, Kavita’s friend, and now a good acquaintance to me. After working a few more hours in the afternoon, Kavita and I officially stopped our work and started our vacation. We went to Suresh and Chaitanya’s place, and saw their two month old daughter Spoorthi. She was fast asleep most of the time, although she did wake up for a few precious minutes. Chaitanya prepared dahi-vada which I can roughly translate to fried split black gram batter in yoghurt. We spent time catching up on the happenings, since my last visit in April, the biggest of which was, of course, the birth of Spoorthi, which in Sanskrit means “inspiration”. We also saw a few Telugu movie songs on Suresh’s newly bought wall projector. It was quite impressive, throwing up a good 4-foot by 8-foot image on the wall. After coming back home, Kavita started dinner preparations. We were expecting Venkat to join us. Venkat is Kavita’s music class colleague and a good friend. Kavita made white brinjal (egg plant) curry and phulkas (that she had learnt from our guest’s mother when she was in town a few weeks ago). Venkat brought some Pulao he had made. We had a wonderful dinner and a lot of good conversation.

We had big plans for Saturday. And after our late dinner the previous evening, an early start was hard, but it was key to pack in all the activities we had planned. And we did manage a good start. Kavita, clad in a sparkling blue saree, and I were heading to the Sri Venkateswara temple in Aurora, a Chicago suburb, by 8:30 AM. We made it is good time and reached temple at 10:30 AM. We confirmed that Praveen, his friend Janaki and the newly wed Kalyan Chakravarthy and Praveena were on their way. They lived in Gurnee and Schaumburg respectively, both Chicago suburbs. We went in and Kavita got the ticket and fruits for the archana. By the time we were done with the archana, Kalyan and KGN had arrived. It was thrilling to see them after a long time. Earlier, I had hardly been standing still or concentrating on the puja in anticipation of meeting my friends. For Kavita, it was the first time she was meeting Praveen, and Janaki was new to both Kavita and me. It was wonderful to meet them. Janaki, effusive in her bubbly, childlike enthusiasm, and carefree wit, and Praveena, calm and confident in her soft spoken intelligence, were great finds. Kalyan and Praveena had just landed back from India, the previous evening and the fact that they could make it to the temple so early next morning was very special.

After the temple, Kavita and I followed Praveen and Janaki to Praveen’s apartment in Gurnee. Praveen’s newly bought Nissan 350Z was an easy car to follow. The gleaming black was never out of sight, and Janaki, who was driving, was a good driver, who always seemed to make moves keeping us in mind. At Praveen’s place, we all changed and headed for the next stop, Lake Geneva, Wisconsin. I drove Janaki’s BMW SUV and we all reached the beautiful town of Lake Geneva around 4PM. Praveen had managed to convince me that ‘wave runners’ were fun. Kavita was not buying any of that. Janaki’s enthusiasm, despite her not knowing how to swim, Praveen’s repeated assurances of its “untoppleability”, the consoling fact that we would have life jackets strapped on, and my seeming confidence in my ability to drive it, finally convinced Kavita. And so we rented them water scooters or call them wave runners. A half an hour wait while the ones on the lake returned was a wonderful opportunity to grab a couple of icecreams at the rustic market place in downtown Lake Geneva. At 4:40 PM, we got our wave runner. After paying close attention to the rules of the lake and operation of the vehicle we launched ourselves into rolling dark blue waters. Kavita was hanging onto my life jacket tight. I was hanging onto the handle bars tighter. A minute into the hour long drive our vehicle dove too hard into a trough and the wave landed itself on us, totally drenching me and partially, Kavita. From then on, we were trying to go very fast only to dry ourselves! The side effect of driving fast was that the vehicle was driving better! It was a lot of fun flying around from wave to wave, with a fountain of water spewing out from the craft’s tail in a tall parabola. Unfortunately we could not measure our speeds, but at times we did seem to go pretty fast. I never dared go full throttle though – maybe next time. On occasions, we also spotted Janaki heralding her craft with unabated aplomb.

After the hour long fun on water, we got back into the SUV and after a short self guided tour of Lake Geneva, headed to the Gurney Mills mall. We did not achieve much shopping there. But as soon as we entered the mall and started looking around, we got confused by the sounds, colors, shapes and spaces. We hunted for a directory/map of the mall, found one, and consulted it. After what seemed like a long time we were on our way to mastering its mysterious patterns. We had figured out where we were on that map. After another few minutes, we figured out where we were trying to go and how to get there. Confidently and cheerfully, we headed off, in the wrong direction, and ended up at the exact opposite end of the “Z” shaped mall. But we did find a few shortcuts to go from one end of the “Z” to another, and we also made sure we walked a good half a mile. A trip to the mall is to me, an opportunity to walk. And the more stuff you buy, the more you walk carrying weights. We must never underestimate the positive effects that malls have on the health of its patrons.

Dinner at Chinese/Thai restaurant, Big Bowl, was great. Janaki’s recommendation of Orange Ginger Ale was duly acknowledged as a wise one, although, the venturesome Praveen tried a vague tasting Pomegranate Ginger Ale, and apparently liked it. After dinner, with Janaki’s hand-written directions in hand, we headed to our hotel near O’Hare, the Holiday Inn Select at Rosemont. With barely the energy to enjoy the view from our 8th floor suite, Kavita and I crashed into the bed.

Sunday was the day of chaos. It was chaos of a good kind. Indecision was rampant, good luck was mixed with bad, anger gave way to refreshing happiness, we were all branded racists and we found sexual discrimination rampant in a famous Chicago landmark.

After the tiring affairs of Saturday, the fact that Sunday started late was no surprise. Kavita and I checked out of the Holiday Inn, drove to Navy Pier on Lake Michigan, and parked our car in a parking deck there for $22.00 for the whole day. Our original plans to rent a bike had to be dropped because of the late start.

Rajat Shah, his wife Smita, and their neighbors Pradeep Reddy and his wife Tania had made it to Chicago the previous day, after a long drive from Troy, Michigan. Rajat was my batch mate at IITG. Smita, I was meeting for the first time, as was Kavita. Pradeep and Tania, we had never met before. Rajat was driving up to Devon Avenue in Chicago after their visit to the Shedd Aquarium. Devon Avenue is where you need to go if you want some great tasting Indian food, especially if you want to enjoy it only after overcoming the travails of finding off-street parking. That was the rendezvous spot for this mini reunion. “Tiffin” was the name of the restaurant where we had decided to congregate at. Rajat picked Kavita and me from Navy Pier. As soon as we found parking on a side street, and started walking, Rajat, who was walking beside me, vanished. He had stopped suddenly and bent down so fast I had to turn around and look for him. He was up in a millisecond displaying in his raised hand, a dollar he found lying on the curb. Smita and Kavita were convinced that it was a fake. Rajat and I insisted it was good. While we were so engaged in arguing about silver strips in currency and other means of authentication, Pradeep showed up with thirty more dollars that he found lying on the road. Since we had already touched these notes before the not so thrilling idea of them being anthrax-laden baits occurred to us, we just decided to pocket the money.

We made it to Tiffin in good time, and on the way we met Praveen and Janaki, who had promised to be there. We also ran into Ashwin Vyas, his wife Tania and their friend Karunakar. The three of them said they would join us at Tiffin for a few minutes. The 8 of us entered Tiffin and promptly asked for a place to seat 10, counting Kalyan and Praveena. We forgot about Vyas, Tania and Karunakar. The reunion was a fantastic experience, in spite of the disorder. Vyas, Tania and Karunakar showed up first. Kalyan, who promised being there in 5 min, walked in after a half hour. And almost an hour after we stepped in we started our buffet. The food was good and the wait staff was accommodating to our dynamic update to the seating requirements. Amidst our congratulating each other on marriages, Vyas’ tips on selecting wine, Rajat’s hearty laughter, Janaki’s cheerful banter, Praveen’s ever smiling countenance, Kalyan and Praveena’s marriage details and loud across-the-length-of-the-table conversations, the food vanished fast. Vyas, Tania and Karunakar had had their lunch at Sabri Nihari, even before we showed up at Devon. So they had joined us for the chit-chat and to draw plans for the remainder of the day. That was the most interesting part of the lunch. Figuring out what everyone was doing next and who would join which group. Kavita and I wanted to go to the Museum of Science and Industry to see the ‘Body World’ exhibit. Praveen and Janaki were planning to go to IKEA, left, came back after a total of 3 minutes and decided to join us to the museum. Kalyan and Praveena were heading to the De Paul Univerity on some personal work. Ashwin, Tania, Rajat, Smita, Pradeep, Tania and Karunakar were going to the downtown in general. So we all started off in different directions. We ran into each other several times since even before we got out of Devon Avenue. The plan was that all of us would try and make it to the architectural boat cruise offered by Wendella cruises, in the evening at around 6:30PM.
Praveen, Janaki, and the two of us, started heading out of Devon, when Praveen spotted a sugarcane juice stall. He jumped out of the SUV even before Janaki realized what happened and bought us sugarcane juices. The juice had a tangy taste to it which Kavita attributed to the jaljeera masala. For $3 a glass the price was somewhat steep, but KGN’s enthusiasm made his treat worth it. Janaki, who originally was not sure if she wanted it, decided to give it a shot.

The disappointment of the day was the fact that the “Body World” exhibit was sold out for the day, and in all probability, for the whole season. Kavita had been looking forward to this exhibit for a long time and was visibly disappointed for a good couple of hours. But we made up for it partially by roaming in and around the museum for a while, watching some video snippets of the exhibit and checking out for the Pioneer Zephyr train exhibit in the atrium. Luckily we had found street parking for free, not too far from the Museum, and therefore were spared of the heartache of buying expensive parking only to not use it. Even if we had gotten in the time constraint due to our planned boat trip would have turned it into a short visit. Yeah, the fox that cant reach the grapes must console itself with the thought that grapes are sour. Janaki stopped at the Millenium Park on the way back, and Kavita and I were promptly huddled out by Praveen, who insisted that we should check out the fountain with a face on it, and the shiny egg. They waited for us on Randolph St. with the blinkers on, while we successfully completed a super quick visit to Millenium Park.

After spending another half an hour to find parking and with the added pressure of making our way to the Wendella Boat Tours’ starting point on time, since Vyas had already bought the four of us tickets, we walked with an added spring. We found Rajat and party, and Vyas and party already at the dock. So we picked up our tickets and joined the queue. Rajat and party apparently had not gotten tickets for the 7 PM tour, and so decided to go up the John Hancock building instead. As I stood in the queue, wondering how many people would fit in that boat, given that the line seemed long, I turned around and saw a familiar, yet unexpected face of Apoorv Saxena! Apoorv was my batch mate and project partner at Purdue University, and lives in San Jose, CA. Seeing him and his girlfriend Ruchika was a pleasant surprise. We caught up on some news after overcoming the initial shock of seeing each other away from our respective bases, and before we got on the boat, which was quite soon. The boat was full on the open top floor and the open front section of the ground floor by the time we all got in, and we got the best seats we could under the circumstances. The cruise that was supposed to last an hour and a half started on a bad note, when we realized that every building that they were talking about in the cruise was on the right hand side. We were sitting on the left, and as such could not see much out of the window. So, the $19 we paid for the ticket was not getting us much for the money. In fact, if anything, it was causing us a lot of grief. So, Praveen, Apoorv, Vyas and I went upstairs to stand and view what was being talked about with such interesting historical references. But promptly a stewardess confronted us with a pleasant but firm, “Sir, for safety reasons, you cannot stand here, and I must request you to go down”. I made an attempt to make her understand that we could not see much from below and that there was no forewarning about this possibility. She offered no help and insisted we go down. I asked her for a refund to which she denied having a say in that matter. So we all walked back down. After another few minutes of trying to follow along, Praveen just got up and said, “I am going up, man”. I followed him. We just went up and stood there. The view was gorgeous and no wonder the architecture tour boat was sold out. But not for long. The stewardess approached us again and repeated her order. Praveen said, “I am not going down”. She tried to tell him her reasons. He said, “Throw me over board, but I am not going down”. He was angry. He was in clear earshot of the fellow passengers and he was loud. The stewardess, in her feigned innocence, wondered what the problem was. And after making it amply clear, Praveen and I saw her walk back. After about 15 seconds, the craft’s captain showed up and gave us the same orders. Praveen did not budge. He looked the tall, young, imposing figure of the captain in the eye and said there is no way he is going down. He made it clear in no uncertain terms that he spent a lot of money, invested a lot of time in this tour and he will not be suppressed into obedience. The captain asked us to talk downstairs since the passengers on the upper floor were hearing all this. We walked down the steps with him and he said he would try to get us a refund after the tour was over. KGN said, “What about all the time I am going to waste on this tour”. So the captain angrily asked us, “So, if I dock right now, will you leave?” KGN said, “Yes”. I agreed. And that was the last we saw of him. The boat which had made a U-turn on the Chicago river and luckily was close to the starting point, started drifting closer to the dock. The four of us got off. Unfortunately, we could not say proper goodbyes to Apoorv, Vyas and others. But as we walked across the plank to the dock, and the lady on the boat’s PA system announced, “We are sorry but we have a small problem and we will be delayed by a few moments. We have a few passengers who want to get off the boat”, we heard approval from the others on the boat in the form of clapping for the stand we had taken. After a few enquiries, we were refunded our money and it was refreshing feeling. Praveen came in for a lot of praise from the three of us.

We were done with the boat cruise and it was still too early for dinner. So we started walking north on Michigan Avenue, looking at the storefronts. We decided to meet up with Rajat and party, who were likely at the John Hancock center. We met them at the base of the John Hancock building. We related our respective experiences, and had a good laugh. We compared the downtown of Detroit to that of Chicago. We talked about the view from the Signature Lounge on the 95th floor of the John Hancock building. Smita informed Kavita that the ladies room on the 95th floor has a big window providing a breathtaking view of the city below. About right then, guy walked up to us and asked for some money. When we all declined, he asked us if we were Christian. We did not respond and continued our chat. He seemed unimpressed and said we were all racists. He said it under his breath and as he hastened away. And probably there was a question in his mind about whether we heard him or not because we all started laughing at being called racists. A black guy calls us brown people racist. We thought that was amusing. After a minute or so he returned to walk right though our group declaring that he wants us all to go back to India and that we were most certainly racist.

After hanging around for a few more minutes and catching up on Smita’s experience with people she thought were muggers in Detroit, and Kavita’s experiences getting lost in downtown Detroit recently, we said our goodbyes. Then the four of us went up to the 95th floor of the building, to the Signature Lounge. The view was great but limited. We decided to not stay for a bite or a drink. Instead we checked out the restrooms. Kavita and Janaki came out elated. The view was apparently spectacular from the large window in there. Praveen and I walked into the Men’s room. Not a square inch of window anywhere. For all we knew we could have been a hundred floor under the surface of the earth. We decided this was not fair. But then as long as the women are happy, we were fine.

We picked up Janaki’s SUV from the parking lot, parked it on the street and stopped for dining at the McDonalds owned, Chipotle’s. The food was good. Praveen’s margarita, served in a no-nonsense plastic glass was not that great. But he was quite drunk, or so he claimed after the single glass. Praveen and Janaki dropped us off at Navy Pier and both of us drove back in just over two hours, recounting the incidents, talking to parents in India, playing crazy-antakshari (a game which Kavita and I play where you sing real or imaginary songs, with accurate or contorted lyrics, in any order you like) and finally reached Normal, exhausted, but very happy with the memorable trip.

Syhamala and Srini also enjoy participating in this kind of a cryptic clue session. Lot of times, since the clues are created on the fly and are not “solidified” yet, they are twisted and mangled as the game proceeds leading to strange guesses and a hearty laugh. Shyamala suggested that I should atleast keep track of these, and therefore I am putting them together here.

Hereditary Clothing
Aim to shop at this store
Mated domesticated
Spherical dance
Direction I threw the stew in
India grows teas in this direction
Direction of old “you”s
Direction prickly things go
Mites multiplication factor
He levitated like this flower
Sue it’s for blowing your nose

The pictures of this trip are here.