Pizza and Panini - Solution

Here is the solution to the puzzle posed in the "Pizza and Panini" column. It is a pretty straightforward puzzle and I have seen variants of it all over the Internet. I remember seeing Martin Gardner pose a version of it in one of his puzzle books as well, IIRC. Anyway, here is the solution.

I am posting this from Las Vegas. "Ask the Delphic Oracle" is here, obviously, to mathematically investigate mathematical games of chance in terms of the mathematics. Pictures to follow - although that might not be for a while.

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(Whiling the days away on Chandrayaan 12) The solution to the puzzle is simply eleven. If the astronauts picked up a total of 11 boots, there could be at most ten of them of one type. The eleventh boot would have to be of the other type.

Elijah the Prophet : a fantasy with riddle

For no particular reason other than the fact that the story has a riddle in the end, here is an extract from a story by Sholem Aleichem.

 An old man with a great gray beard down to his knees. An old face, yellow, wrinkled endlessly, fine and good. And eyes-such eyes. Good tender friendly loving and faithful eyes. Stooped over a great, great cane with a sack on his shoulders-and sha shtill, he comes wordlessly straight to me.

“Nu, yingeleh get into the sack,” says the old man to me so softly and sweetly.

I ask him: “To where?” He replies: “You’ll see afterwards.” I don’t want to go.” He tells me again. I ask him: “How can I go with you? I’m a rich man’s son.” Says he: “So you’re a rich man’s son, what yichus [family connection] is that? By me, you’re not an only son.” Say I: “Fussed over, from seven the sole survivor. They’ll find out that I’m gone and they’ll not be able to bear it. They’ll die, especially Mama.” He looks at me, the old man, softly and sweetly like earlier: “If you don’t come with me, sleep well, but sleep forever.” I begin to cry: “Does that mean that I will die? They’ll not be able to endure it, especially Mama.” “You don’t want to die? Then come with me. Separate from your parents and come.” “What do you mean? How can I go? I’m an only son, from seven the sole survivor.” He speaks up more strongly to me: “For the last time, yingel, choose one of the two: either separate forever from your parents and come with me or remain here and sleep forever. Forever.”

When he finished these words he took a step away from me and turned to the door. What should I do? Go with the old man God knows where, to oblivion–and my parents would die? An only son, from seven the sole survivor? Or remain here and sleep forever? That means that I myself would die. I hold out to him both my hands with tears in my eyes: “Elyohu HaNovi, good, loving Elyohu, give me a moment to think.” He turns to me his fine old yellow wrinkled face with the great gray beard down to his knees. He looks at me with his fine good loving faithful eyes and gives me a smile: “One minute I give you to think, my child, but no more.”

The old man leans on his great, great cane and waits.

The question is: what could I devise in that minute so that I needn’t have to go with the old man or sleep forever. Ah, nu, who can guess?

Update (September 10): A correction : it isn't quite true that this story was posted for no particular reason. I sent the piece below "Now, I don't want to sound like a politician ..." to a professor and he said that this piece reminded him of Sholem Aleichem. It is the sort of stuff I was aiming for. It is not that I had Sholem Aleichem specifically in mind, but it is done in the same folksy sort of a way and it is, like, you know, kind of intended to talk about everyday people and their concerns, but it is also designed to satirize larger things. So, color me pleased.

Puzzle contest announcement

A question that just came in regarding the Column 3 puzzle: "Does the astronaut take out the boots one at a time or does she take out k boots all at once (where 0 < k <= 20)?" It is the latter. The astronaut takes out k boots all at once (where 0<k<=20). The question basically is - what is the minimum k such that she is guaranteed to have a pair of boots - one left and one right?

One could easily change the puzzle in subtle ways and pose variations of the same question, and one would get vastly different answers depending on the mathematics of the problem. And that is the beauty of it.

Anyway, the question came in in the context of the announcement of a contest for the puzzle in Column 3. We are running a contest for this puzzle in association with the Stanford Math Circle. The problem is really quite a bit easier than it looks. The prize is a $40 Amazon.com gift card. Anybody is eligible to participate but you have to pick up your prize at the Stanford Math Circle meeting. If you are not from the Bay Area, you could use a proxy to pick up the prize. That's about it about the puzzle contest. Happy solving!

Pizza and Panini

Indiatimes' main page is carrying an edited version of the latest installment of our mathematics column entitled "Pizza and Panini". It is written in the style of the New Yorker. Hope you enjoyed reading it as much as we enjoyed writing it. (For those who came in late, Chandrayaan-12 is the rocket on which Anand and Ravi are traveling.) Please send in your answers to the puzzle in the column to the following email address : askthedelphicoracle@gmail.com. Happy solving!

PIZZA AND PANINI

This article is in collaboration with Prof. Krishnan Shankar, Professor of Mathematics at the University of Oklahoma.

The Oracle Asks

The Sanskrit grammarian Panini is at his friend Socrates’ place in Athens.

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Boy. Here is the tea.

Socrates. Thank you.

Panini. The boy, he understands Greek Mathematics, does he not?



Socrates. Yes, indeed; he was born in the house.



Panini. Can you talk to him about mathematics?

Soc. Certainly. Attend now to the questions which I ask him, and observe whether he learns of me or only remembers.

 

Pizza and Panini

The third installment of our column is coming up at the Times of India. And to give you an update on the TEDx talk - my preparations for the TEDx talk are coming along well. Tomorrow, I am scheduled to talk to Nori Gerardo Lietz, an expert on world real estate markets here at Stanford. So exciting!

I expect to have an interesting talk with her and I imagine she would have some very interesting things to say on the topic, which <drumroll> is "Why there are cows on the street in India?" Anyway, the latest installment of our column is almost there. Hope you enjoy this month's puzzle.

Einsteinian relativity, Milton Friedman, Steven Pinker, the Socratic method et cetera

I would like to make a few additional points regarding the wires puzzle (a.k.a. the Chandrayaan Engine Room puzzle) in the column, and then switch over to some thoughts from an organizational perspective. The wires puzzle is an old one. There are versions of it all over the Internet. What is different here is the use of the Chandrayaan theme. Since this is happening aboard Chandrayaan, there could be relativistic effects that may need to be considered. The point of posting this puzzle using the Chandrayaan theme is to say that, sometimes, old problems can be looked at in new ways.

That is, one interesting way to think about the problem is to ask if the solution to the problem changes if we have relativistic effects. If the rate at which the wires burn changes over time, then you suddenly have a more interesting problem. The point is that if we shift the 'frame' of the problem, then sometimes, the problem itself changes. (See Steven Pinker's discussion of some of these 'framing' effects in Chapter 5 of his book "How the Mind Works".)

We are planning on a puzzle in Finance in a future column. With Finance problems, there may be other assumptions such as time preference of money. Milton Friedman talked about problems and assumptions in his famous paper "The Methodology of Positive Economics". As Friedman put it, theories almost always make assumptions, and so the solving of a puzzle is a Socratic process in that the framer of the problem and the anwerer must agree on a certain set of assumptions. But beyond agreeing on the set of assumptions, nothing more is required.

Now, puzzles are often used in interviews, and one person even termed his interviewing process Socratic. I do not think, however, that a Socratic approach to puzzles is a good one. It seems to be a rather mistaken approach. From what I have seen, the dialogue that ensues after a puzzle is proposed is almost never about clarifying assumptions behind the problem. The dialogue is about the interviewer providing the interviewee subtle clues to solving it. How many clues the interviewer provides the interviewee depends on how much the interviewer likes the interviewee at first blush. And that's not ideal if we want the organization to pick the best people since first impressions can often be mistaken. There is a problem here that I am calling your attention to and it is an organizational one. Most interview puzzles are quite sufficiently specified, and so the requirement these days that interviewers engage in a dialogue with the interviewee seems rather wrong headed. It ought to be sufficient for the interviewer to clarify assumptions underlying puzzles, and then let the interviewee figure out the rest on his or her own.

The whole point of puzzles is to have an objective means of analyzing candidates. There is much subjectivity in almost every other part of the process. Companies really ought to change the way they interview candidates. Why companies continue to do interviews they way they do is, of course, a whole different ball of wax.

Update (June 18): This is a very technical post. You can safely skip this post and still enjoy the puzzles in this column. You can also assume that there are no relativity effects in coming up with a solution. 

The two wires puzzle a.k.a. the Chandrayaan engine room puzzle

A friend of mine had a question on this month's main puzzle, and so I would like to make one clarification regarding the two wires puzzle. The rate of burning of the two wires may be termed as "fixed but unknown". The first wire may burn as follows : 1 minute for the first 1%, another 2 minutes for the next 1 % and so on. At the end of 10 minutes, only 9% of the wire may have burned. However, since each wire takes an hour to burn through, the rest of the 91% of the first wire will burn in the next 50 minutes. The wire may be burned from either end. Similarly for the second wire.

Note that the way that the second wire burns may not be the same as the way that the first wire burns. Thus, the second wire may burn as follows : 2 minutes for the first 1%, 4 minutes for the next 1% and so on. At the end of 10 minutes, only 3% of the wire may have burned. But since the wire takes an hour to burn through, the rest of the 97% of the second wire would burn in the next 50 minutes. I use the term "fixed but unknown" because while the way the two wires burn is not known, the way the two wires burn does not change over the period of time in question.

(Chandrayaan Engine Room) Chandrayaan-12 has run into trouble. The problem is in the engine room, and the initial investigation into the problem has revealed that the positron motor needs to be restarted. This needs to be done exactly 45 minutes after the neutrino drive is turned off. However, the clocks in Chandrayaan are no longer reliable. All you have are two wires. The two wires each take exactly an hour to burn. They don't burn uniformly, however. So, for instance, the first half of the first wire may take 13 minutes to burn and the second half 47 minutes. Is it possible to measure out exactly 45 minutes using the two wires? If so, how?

Update: Mathematically speaking, let the length of the first wire that burns in time t be f1(t). This function is not known. It could be of the form [ f1(t) = k.t ], but it could be quite different as well. You don't need to understand any advanced mathematics to solve the problem, however, and for this reason, we have avoided using mathematical notation for the problem.

Inequality of the means

To skip the talk and go straight to the this month's main puzzle, just go to the Indiatimes article (linked here) and scroll all the way down.

Indiatimes is running an edited version of this article, which is the next installment of the puzzle column. This article is in collaboration with Prof. Krishnan Shankar of the University of Oklahoma.

The main problem in this article, the geometric version of the Inequality of the Means, has been chosen for its simplicity and elegance. It is one of those mathematical problems that is easy to state but ridiculously hard to prove. There are two puzzles this month. Please be sure to use the two different Subject lines mentioned in the article to help us distinguish which puzzle it is you are replying to.

A note regarding the "New Dice" puzzle - the problem is not asking whether it is possible to number the dice such that each possible outcome 2 through 12 occurs with equal probability. The problem is to come up with numbers a1, a2, a3, a4, a5 and a6 for one dice and b1, b2, b3, b4, b5 and b6 for the second dice (together, a1 through a6 and b1 through b6 form a "numbering") such that the following properties are satisfied:

• Together, a1 through a6 and b1 through b6 constitute two sets of 1 through 6.
• The "numbering" is not identical to the default "numbering". 
• The "numbering" would lead to the same probability distribution of outcomes as the 1,2,3,4,5,6; 1,2,3,4,5,6 numbering.

One possible "numbering" could be : 1,1,2,2,3 and 3 for Dice 1; and 4,4,5,5,6 and 6 for Dice 2. The only problem is that this numbering would not lead to the same probability distribution as the default 1,2,3,4,5,6; 1,2,3,4,5,6 numbering. So, without further ado, the column is below.

INEQUALITY OF THE MEANS

This article is in collaboration with Prof. Krishnan Shankar, Professor of Mathematics at the University of Oklahoma.

The Oracle Asks

This article’s material came forth from the fertile mind of the extraordinary John Conway. References to the mathematics of the problem are listed at the end. It was a pleasure to discuss this problem with Prof. Dror Bar-Natan. Prof. Bar-Natan’s exposition and Javascript applet make the subject come alive on his website (http://www.math.toronto.edu/~drorbn/), which is certainly worth a visit.

We start with a well-known inequality from high school algebra: let a and b be any two non-negative numbers. Then, their arithmetic mean is at least as large as their geometric mean, i.e.,

square_root(a *b ) <= ((a + b)/2)

Equality occurs precisely when a = b. This is not hard to prove algebraically, but here is a nice geometric proof. Consider the following figure where a square of side length a + b encloses 8 right angled triangles of orthogonal sides a and b each. 

Oklahoma Sooners

We have written up columns for the next few months, and will be proof reading them and preparing them for publication over the next few weeks. We will get it all done real soon. For the summer, we are planning to have three short columns, ones that will mainly consist of just the puzzle itself. The first summer puzzle has a beach theme, and the second one a game theme. The third is still work in progress.

While we are on the topic of games, it would be good to say a little something about Oklahoma Sooner football and even American football games in general. A football game in Norman or Austin is much more than a game. It is a celebration. The long running rivalry between the University of Oklahoma-Norman and the University of Nebraska-Lincoln no longer exists, since it has now been replaced with the Red River Rivalry, but when it was around, it was one for the ages. Here is a clip from a 1986 game.

   

Even people from Europe typically don't realize how big these football games are as events. A game between these two teams from relatively small towns, Lincoln and Norman, the former with a population of about 250,000 and the latter with about 100,000, is quite a spectacle even on TV. Being a part of the spectacle was eye opening for me when I was in America fresh from India. To me, it was proof of the robustness of the American economy that these relatively small towns are able to stage high attendance games through the football season and yet have all the pomp and pageantry that you might only expect to find in games having large metropolitan area audiences. The rivalry game is often the most closely fought one in the football season, and all the big universities have a rivalry game. In the big rivalry game, the wins are to be cherished and celebrated and the losses are never to be forgotten, decades to come. You really have to see it to believe it.

Ask the Delphic Oracle

To skip the talk and go straight to this month's main puzzle, just click on the Indiatimes article (linked here) and scroll all the way down.

Indiatimes' main page carried an edited version of this article on Operations Management and the applications of mathematical analytics there.

Ask the Delphic Oracle

“Ask the Delphic Oracle” is a new column in the Times of India. As part of this column, we plan to run a new puzzle every month. We will allow three to four weeks for you to solve the puzzle. Please write in with your answers to: askthedelphicoracle@gmail.com. We will publish the names of the people who answered the puzzle correctly (randomly chosen out of the first fifty). Good luck!

Ask The Oracle:

Q. I am an Australian in California. I have noticed that a lot of Indians here drive Toyotas. Why do so many Indians drive Toyotas?

Answer. While we put our business analyst hats on, may we point out that there are excellent reasons to own a Toyota? The main reason is, of course, the quality of the car. But how is Toyota able to produce cars of such high quality? Behind the answer to this question lies the story of the machine that changed the world.

Before there were Hondas and Toyotas, there were Fords. The big idea that Henry Ford came up with was that of the assembly line. Henry Ford realized that if you organize a car factory floor like a meat packing assembly line where each worker gets to specialize on one piece of the job, then the productive efficiency dramatically increases. From this was born the modern automobile assembly operations setup, the machine that changed the world.  The Ford automobile assembly operations setup was further improved upon by the Toyota Motor Company by means of the Toyota Production System. The Toyota Production System consists of a unique combination of social and technical processes that makes it possible for them to create very high quality cars with low rates of failure. This makes Toyotas cheap to own in terms of total cost of ownership and easy to maintain, but this is clear only after you have been educated on many different aspects of the matter of car ownership. Although Toyotas are expensive to buy, they pay off in the long term, and have low total cost of ownership. It is not surprising then that Indians in America, given their high level of price sensitivity, like to own Toyotas.

The Oracle Asks:

Why are Toyota cars of such good quality? Why are shipping containers sometimes sent halfway across the world half full? Why do clothing stores such as Pantaloon and J. C. Penny have so many extra trousers sitting around on shelves? If the average expected sales of iPads is 100 units per month, does it make sense for a store to have more than a hundred tablets in stock? These and many other questions may be answered using operations management techniques.