Opinion | A constitution enacted, the kangaroo way6 min read . Updated: 20 Nov 2020, 01:23 PM IST
- With professor Martin Kruskal’s trick, all paths lead to the same word in a passage
The other day, as part of the ongoing Tata Litlive festival, I spent a while watching Shashi Tharoor and Christophe Jaffrelot discussing ideas of patriotism and nationalism. Especially given the times we live in, there were many interesting themes that came up. One that stayed with me was the need to spread knowledge of and an appreciation for our Republic’s founding document, the Constitution, far and wide. Farther and wider, in fact, than we have managed in our 70-plus years.
Doing so, Tharoor and Jaffrelot argued, is not just about showing Indians that this document spells out matters that are in our interest, that protect us and give us rights. It’s also about building a “civic nationalism" that transcends far narrower calls to religion, caste, language or other identities we hear regularly today.
This made admirable sense to me. So I thought, what can a mathematics columnist do to aid this pursuit? Well, one thing I can do is get you to read the Preamble to the Constitution. While you do, I can seek to persuade you, in a mathematical way, that it was indeed enacted for us all. I mean exactly that: enacted.
Let’s jump right in, why not? Here’s the Preamble:
“We, the people of India, having solemnly resolved to constitute India into a sovereign socialist secular democratic republic and to secure to all its citizens
Justice, social, economic and political;
Liberty of thought, expression, belief, faith and worship;
Equality of status and of opportunity;
and to promote among them all Fraternity assuring the dignity of the individual and the unity and integrity of the Nation;
In our Constituent Assembly this 26th day of November 1949, do hereby adopt, ENACT and give to ourselves this Constitution."
To remind you, I’ve put one word above in upper-case: “ENACT".
Here’s what I want you to do. Pick any word from among the first few words (more or less the first line, or the beginning, of the Preamble). Count the number of letters in your word. Move forward that many words. Count the number of letters in the word you landed on. Move forward that many words. Keep going like that. (If you land on either, treat “26th" and “1949" as four-letter words.)
For example, suppose you pick the third word, “people". That’s six letters long, so jump forward six words and arrive at the ninth word, “to". That’s two letters, so skip forward to “India". Forward five words to “secular" ... and on like that.
Pretty straightforward? Well, think of this: Whichever word you choose as your starter, you will arrive at “ENACT". Go ahead, try it, I am not making this up. “People" as above will take you to “ENACT". The first “India", for a second example, takes us to “constitute", “to", “to", “its", “social" ... and eventually to “ENACT". The same with “solemnly" and “resolved".
Baffled? Delighted? Did our founding fathers, deliberating in their Constituent Assembly, carefully craft the Preamble so that this would happen? Are you imagining them dreaming up and then placing words like “to" and “solemnly", “people" and “its", so that this little bit of magic would ensue? So that future citizens of India would be reminded of how we ENACTed and gave ourselves our Constitution? Is this meant to be a subtle hint to generations of Indians to come, to read and take pride in our Constitution?
You can certainly see it in all those ways. But the truth is no, the members of the Constituent Assembly did not deliberately design the Preamble this way. Actually, they had no idea this would happen. That is, this inexorable arrival at “ENACT" was not planned at all.
And yet it is also no accident. What I mean is, you can take nearly any passage to conduct this little exercise on, and you will find there’s a particular word which every path takes you to. (Actually, I should say “nearly" every path, but I won’t get into that here). One of my mathematical heroes, John Allen Paulos, has a short video in which he uses a passage from the American Declaration of Independence. “Does the Declaration of Independence Guarantee Happiness?", he calls the video, because in that case all word-hopping routes lead to “HAPPINESS".
This delightful trick is a variation on the “Kruskal Count", a card trick invented by the Princeton University professor Martin Kruskal (1925-2006). Kruskal had plenty of interests, among them an abiding one in black holes, to the theory of which he made significant contributions. But he was also a congenitally playful mathematician, and his Kruskal Count is a testament to that side of him.
With cards, Kruskal’s trick goes like this: deal a full deck face-up in rows on a table. Take the value of the first card and move that many cards to the right (aces are valued at 1, and Kings, Queens and Jacks at 5). Take the value of the card you’ve landed on and jump that many to the right, and so on. When you’ve reached the last possible card in this path—meaning its value will take you past the end of the deck—turn it over. Now ask one of your adoring friends to start at any one of the first ten cards—though not the first, which you started on—and follow the same jumping procedure. About 80% of the time, she will finish on the same card you’ve turned over.
So just what is going on here? Here’s why it works. Whatever the card your adoring friend starts on, it could have been one you landed on —because each step in this process advanced you by between 1 and 10 cards. If it is one of yours, she’s clearly going to follow your path, and we’re done. But if it isn’t, how likely is it that she lands on one of your cards as she keeps moving? With every move she makes, there’s at least a 1/13 chance she will land on a card you landed on. That’s because there are 13 different kinds of cards the one she’s on could be (ace to 10, J, Q and K); and it’s “at least" because four of those cards move her ahead 5 steps (5, J, Q and K). So there’s a 4/13 chance she moves by 5, but a 1/13 chance she moves by some other amount.
That is, each time she jumps, there’s at most a 12/13 chance she does not land on a card you touched down on. With every move, that gets compounded, and pretty soon it’s more likely than not that she will arrive at one of your cards, which means she will end her journey on your final overturned card. Take a bow: your trick worked like a charm.
Incidentally, Kruskal’s Count is more than just a clever trick. The British mathematician John Pollard came up with the same idea on his own and applied it to certain cryptographic techniques. It’s known there, in fond tribute to the jumps, as “Pollard’s Kangaroo Method". Two other mathematicians, Ravi Montenegro and Prasad Tetali, even used his method in a paper with the delicious title “How Long Does it Take to Catch a Wild Kangaroo?" (Proceedings of 41st ACM Symposium on Theory of Computing, 2009).
As for a random passage of text, like the American Declaration of Independence or the Preamble to our Constitution, the reasoning is much the same. So while our founding fathers did not plan the Constitution mathematically, they certainly put in a lot of hard work before it was ENACTed.
So please READ it.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun