Tag Archives: mathematics

Evolution of Language

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We know that statistics (which is different from mathematics) plays an important role in various other sciences (mathematics is not a science, it’s an art). But still I would like to discuss one very interesting application to linguistics. Consider the following two excerpts from an article by Bob Holmes:

1. ….The researchers were able to mathematically predict the likely “mutation rate” for each word, based on its frequency. The most frequently used words, they predict, are likely to remain stable for over 10,000 years, making these cultural artifacts, or “memes”, more stable than some genes…..

2. ….The most frequently used verbs (such as “be”, “have”, “come”, “go” and “take”) remained irregular. The less often a verb is used, the more likely it was to have been regularised. Of the rarest verbs in their list, including “bide”, “delve”, “hew”, “snip” and “wreak”, 91% have regularised over the past 1200 years…….

The first paragraph refers to  the work done by evolutionary biologist Mark Pagel and his colleagues at the University of Reading, UK. Also, “mathematically predicted” refers to the results of the statistical model analysing the frequency of use of words used to express 200 different meanings in 87 different languages. They found the more frequently the meaning is used in speech, the less change in the words used to express it.

The second paragraph refers to the work done by Erez Lieberman, Jean-Baptiste Michel and others at Harvard University, USA.  All people in this group have mathematical training.

I found this article interesting since I never expected biologists and mathematicians spending time on understanding evolution of language and publishing the findings in Nature journal. But this reminds me of the frequency analysis technique used in cryptanalysis:

Building Mathematics

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Let’s talk about the work of a mathematician. When I was young (before highschool), I used to believe that anyone capable of using mathematics is a mathematician. The reason behind this was that being a mathematician was not a job for people like Brahmagupta, Aryabhatta, Fermat, Ramanujan (the names I knew when I was young). So by that definition, even a shopkeeper was a mathematician. And hence I had no interest in becoming a mathematician.

Then, during highschool, I came to know about the mathematics olympiad and was fascinated by the “easy to state but difficult to solve” problems from geometry, combinatorics, arithmetic and algebra (thanks to AMTIVipul Naik and Sai Krishna Deep) . I practiced many problems in hope to appear for the exam once in my life. But that day never came (due to bad education system of my state) and I switched to physics, just because there was lot of hype about how interesting our nature is (thanks to Walter Lewin). 

In senior school I realised that I can’t do physics, I simply don’t like the thought process behind physics (thanks to Feynman). And luckily, around the same time, came to know what mathematicians do (thanks to Uncle Paul). Mathematicians “create new maths”. They may contribute according to their capabilities, but no contribution is negligible. There are two kinds of mathematicians, one who define new objects (I call them problem creators) and others who simplify the existing theories by adding details (I call them problem solvers). You may wonder that while solving a problem one may create bigger maths problems, and vice versa, but I am talking about the general ideologies. What I am trying to express, is similar to what people want to say by telling that logic is a small branch of mathematics (whereas I love maths just for its logical arguments).

A few months before I had to join college (in 2014), I decided to become a mathematician. Hence I joined a research institute (clearly not the best one in my country, but my concern was just to be able to learn as much maths as possible).  Now I am learning lots of advanced (still old) maths (thanks to Sagar SrivastavaJyotiraditya Singh and my teachers) and trying to make a place for myself, to be able to call myself a mathematician some day.

I find all this very funny. When I was young, I used to think that anyone could become a mathematician and there was nothing special about it. But now I everyday have to prove myself to others so that they give me a chance to become a mathematician. Clearly, I am not a genius like all the people I named above (or even close to them) but I still want to create some new maths either in form of a solution to a problem or foundations of new theory and call myself a mathematician. I don’t want it to end up like my maths olympiad dream.

Teaching Mathematics

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One of the most challenging and rewarding thing associated with being a math enthusiast (a.k.a. mathematician) is an opportunity to share your knowledge about the not so obvious truths of mathematics. A couple of years ago, I tried to communicate that feeling through an article for high school students.

When I joined college, I tried to teach mathematics to some kids from financially not-so strong family. Since they had no exposure to mathematics, I had to start with  concepts like addition and multiplication of numbers. My experience can be summarized as the following stand-up comedy performance by Naveen Richard:

After trying for about a couple of months to teach elementary mathematics, I gave up and now I discuss mathematics only above the high school level. Last week I delivered a lecture discussing the proof of Poncelet’s Closure Theorem:

Whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

I had spent sufficient time preparing the lecture, and believed that I was aware of all possible consequences of this theorem. But, almost half way through the lecture one person (Haresh) from the audience of 10 people, pointed out following fascinating consequence of the theorem:

If an n-sided polygon is inscribed in one conic section and circumscribed by the other one, then it must be a convex polygon and no other m-sided polygon (with m≠n) can be inscribed and circumscribed by this pair of conic sections.

This kind of insights by audience motivates me to discuss mathematics with others!

Revision 1: Inquisitive Mathematical Thinking

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In this post I wish to expand my understanding about, “asking Why?“.

In 1930, David Hilbert gave radio address lecture. I want to discuss following paragraph from that lecture (when translated to English):

With astonishing sharpness, the great mathematician POINCARÉ once attacked TOLSTOY, who had suggested that pursuing “science for science’s sake” is foolish. The achievements of industry, for example, would never have seen the light of day had the practical-minded existed alone and had not these advances been pursued by disinterested fools.

Science exists because we (human beings) want to find reason for everything happening around us (like how air molecules interact, which bacteria is harmful…) . We claim that this will enrich our understanding of the nature thus enabling us to make rational decisions (like when should I invest my money in stock market, from how much hight I can jump without hurting myself…).

Let me illustrate the point I want to make: Mathematicians make observations about real/abstract objects (shape of universe/klein bottle) and try to explain them using logical arguments based on some accepted truths (axioms/postulates). But today we have “science” for almost every academic discipline possible. Therefore, we (human beings) have become so much obsessed with finding reasons for everything that we even want to know why the things happened a moment ago so that we are able to predict what will happen in a moment from now. So the question is:

Should there be a reason for everything?

Can’t some thing just be happening around us for no reason. Why we try to model everything using psedo-randomness and try to extract a meaning from it? In case you are thinking that probability helps us understanding purely random events, you are wrong. We assume events to be purely random, we are never sure of their randomness and based on this assumption we determine chances of that event to happen which infact tells nothing about future (like an event with 85% chances of happening may not happen in next trial).

In same spirit, I can ask: “Should there be reason for you being victim of a terrorist attack?” We can surely track down a chain of past events (and even the bio-chemical pathways) leading to the attack and you being a victim of it.

Why we try to give “luck” as reason for some events? Is this our way of acknowledging randomness or our inability to find reason?

Moreover, David Hilbert ends his lecture with following slogan (in German):

Wir müssen wissen, Wir werden wissen.

which  when translated to English means: “We must know, we will know.”.

Hyperbolic Plane Example

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Few months ago I gave a lecture on Non-euclidean geometry and it was a bit difficult for me to give audience an example of hyperbolic surface from their day-to-day life. While reading Donal O’ Shea’s book on Poincaré Conjecture I came across following interesting example on pp. 97 :

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Negatively curved cloth will drape a woman’s side (© Donal O’ Shea, 2007)

Estrogen causes fat to be stored in the buttocks, thighs, and hips in women. Thus females generally have relatively narrow waists and large buttocks, and this along with wide hips make for a wider hip section and a lower waist-hip ratio compared to men. The saddle-shaped area on a woman’s side above her hip has negative curvature.

One can imagine cloth (it is flexible but does not stretch, hence an isometry) that would drape it perfectly. Here the region inside a circle of given radius contains more material than the same circle on the plane, and to make the cloth the tailor might start with a flat piece of fabric, make a cut as if he/she were going to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric or a gusset. Negatively curved cloth would have lots of folds if one tried to lay it flat in  dresser.

If one tries to extend a cloth with constant positive curvature (like a cap), in all directions, it would close up, making a sphere. On the other hand, if one imagines extending a cloth with constant negative curvature in all directions, one gets a surface called hyperbolic plane.

Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines.

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Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point).

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Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics.

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P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“.

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Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics.

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We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.

Mathematics Today…

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When we talk about doing mathematics, what comes to  our mind is Blackboard-chalk, notebook-pen and books. No doubt that these are and will remain one one of the most important instruments leading to elegant mathematical discoveries.  But, the evolution of technology we use has also affected the way we do, learn and share mathematics. In ancient time, mathematics was shared in form of books and letters. Then in 17th Century people started publishing academic journals periodically, which has today become one the most profitable business (like pharmaceuticals).  In 1960s computer algebra systems were invented (called MATHLAB). Then in 1970s books were digitized and today we have dedicated ebook readers.  Another major challenge  of publishing mathematical knowledge was to be able to typeset weird symbols, and this problem was fully solved using computers in 1978 by Donald Knuth. (to know more about this transition read this discussion).

Now it’s 21st century and the shape of sharing mathematical knowledge has changed significantly in past decade.  To begin with, in 2003 Poincaré conjecture’s solution was not published in any journal but was rather posted on arXiv. Today we have people on social networking sites like Facebook, Twitter, Google+, Tumblr, Weblogs...  who let you know the results just as they are being cooked up. For example, Live-tweet of Babai’s first Graph Isomorphism talk, in this talk one of the most interesting theorem of 2015 was proved. Many big shots announce their big results directly on their Weblogs, for example Terence Tao announced his proof of Erdős Discrepancy Problem on his blog. Today we can have interactive textbooks (like this), articles (like this) and assignments (like this) with advent of MathJax, SageMath

So far I have been concerned about “print” mathematics, but with advent of cheap internet, whole new methods of mathematical ideas sharing have come into picture. Today almost every reputed research organization maintain video lecture archives (IAS, CIRM,  IHÉSIHPInstitut Fourier, MatScience). Apart from mainstream mathematics, popularization of mathematics has become much more interesting today. We have lots of interesting mathematics popularization channels on YouTube like ViHartNumberphile, Mathologer, 3Blue1Brown, The Global Math Project,… and SoundCloud like BBC Radio 4: More or Less, ACMEScience ,… For a big-list of online mathematics videos see this and for big-list of mathematics podcasts see this.

Before this internet era, there were similar mathematics popularization attempts. Like my favourite: “Donald Duck in Mathmagic Land” (1959) [updated the link on 28 Dec’18]

But I’m not aware of existence of mathematical radio programs back then. So, if you know about such radio programs,o please let me know about them as comments below.

135th Carnival of Mathematics

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Welcome to the carnival…

Before we begin, let’s know our number-friend of this month. 135 is the smallest three-digit number that is the sum of its first digit and the square of its second digit and the cube of its third digit: \displaystyle{135 = 1 + 3^2 + 5^3}.

Now, get hold of pen and paper, the carnival begins…

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Cross-Number Puzzle – Numbrcise.com

This is a “medium” brainteaser to help you warm up. To play, you must fill in all the blank squares in the grid with numbers ranging from 1 to 9 in order to find a synchronized solution to a series of horizontal and vertical equations – all at once.


Three dimensional tessellation of crosses – circlesandtriangles (Dan)

Day after tomorrow is 118th birthday of Maurits Cornelis Escher and will be celebrated as first “Tessellation Day” [details]. And in this article, Dan investigates the three dimensional analogue of Greek Cross tessellation using animations. He elegantly explains the mathematics involved behind tessellation by reducing  the 3D problem to a 2D one.

Snell and Escher – Joshua Bowman

When we study the concept of refraction, one of the central law is “Snell’s Law”. Motivated by Johann Bernoulli’s (a mathematician) clever application of Snell’s Law to solve brachistochrone problem,  Joshua introduces reader to hyperbolic geometry via works of Maurits Cornelis Escher  (an artist) using Snell’s Law! I have many times discussed hyperbolic geometry with young  minds, but this introduction is really illuminating.

5 Mistakes to Avoid when Drawing a Soccer Ball – David Swart

At some point of time, you must have tried your hand at drawing various objects around you. Moreover, soccer is claimed to be most played sports in the world. But still many of us can’t draw a soccer ball properly! In this lovely article, David helps us to mathematically identify and correct these mistakes…


Boolean Algebra – mathsbyagirl

We all use electronic devices and they  will not work without the transistors linked together on a silicon chip. These transistors interact with each other using “on” or “off” state. This interaction can be mathematically modelled using “Boolean Algebra”. In this article, the author has successfully transmitted her passion and excitement for the subject to other people. This article is accessible to anyone curious enough to learn about this subject.

Banach-Tarski Paradox  –  Austin Lawson

If you try to take a stroll through “mathematical logic”, you will encounter the statements that contradict themselves and yet might be true.  Such statements are called paradoxes. In this article, Austin gives a sketch of the Banach-Tarski theorem and paradox.  It is intended for mathematicians of intermediate skills though the first few paragraphs are accessible to more people.


Programming is a piece of cake  – Paula Rowinska

This is an article discussing the importance of computing in the modern maths, especially the applied areas. There’s a common misconception that mathematicians work only on very pure and useless for the society problems – Paula explains that it’s a myth.

How to assign partial credit on an exam of true-false questions? Terence Tao

Many of us would have appeared for exams in May, and hence this article by a gifted mathematician becomes oddly relevant! In this article Prof. Terence does a serious analysis of grading system using techniques from Probability and Statistics. In case you happen to be a person associated with academia, this is worth reading.


Float like butterfly and Sting like a Mathematician!  – Nira Chamberlain

Muhammed Ali has been regarded as the most influential sportsman of the century. A week ago, he died aged 74. This is intriguing story about how a heavyweight boxing champion influenced Dr. Nira to become a mathematician.

Interview with a mathematician: Maria Chudnovsky  – Anthony Bonato

Graph Theory is one of those branches of Mathematics which consist of simple to state but difficult to solve problems. In case you are not familiar with it, just spend some time on this “Math for seven-year-olds” kit. Prof. Anthony interviewed Maria Chudnovsky, a leading mathematician specializing in graph theory. Maria is famous for proving the “Strong Perfect Graph theorem”, which was open for forty years. Maria is engaging and gives great advice to young mathematicians. She also talks about her upbringing, including facts you can’t find on her wiki page or other interviews.


Prime After Prime  – Brian Hayes

There’s lots about the prime numbers that seems random; you can even play a good game of dice with them. But in March Robert J. Lemke Oliver and Kannan Soundararajan discovered some remarkable biases or correlations between pairs of consecutive primes. Brain explores this discovery in computer code and “very illuminating” pictures.

Chi-square goodness of fit test example and prime numbers – John Cook

This article builds upon Brian Hayes‘ article about the distribution of primes, which we just discussed.  Motivated by the argument that “Primes aren’t random, but sometimes it can be useful to treat them as if they were.”, John writes about the connection between statistics and number theory. Mathematically mature audience will surely enjoy reading this.


Maximal density subsquare-free arrangements – Peter Karpov

I will end this Carnival with an open problem. In this article, Peter discusses some computational results for the “no subsquares problem”. The statement of problem is as follows

What is the largest number of points that can be placed on a N × N grid so that no four of them form a square?

Get your hands dirty and try to find an efficient algorithm to calculate answer for N>16!

 


 

But, before we say good bye to this edition of monthly roundup, let me remind you to visit the previous edition which was hosted by Kartik at Comfortably Numbered. Join us next time for the 136th edition, hosted by Manan at Math Misery. You can find all the other Carnivals, and submit articles for future carnivals at: http://aperiodical.com/carnival-of-mathematics/ 

Why I want to be a Mathematician?

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A few weeks ago I revised my views about “Why I love Mathematics?“. It has been two years since I have been trying to get into mainstream mathematics research. It will take another four to five years for me to start contributing to mainstream mathematics research. In this post I will try to add a bit more to what I wrote an year ago.

Though mathematics is the main thing my life revolves around,  it’s not the only thing. I love doing few other non-mathematical things. Few months ago I watched the movie “Ship Of Theseus” by Anand Gandhi , and it is one of the few movies I have agreed to watch twice. In case you are curious to know about Theseus’ paradox, I will suggest this ted-ed video

As you may be aware of, there are a lot of people who are not mainstream mathematicians (i.e. working in a Research Organization in a specific research area) but still claim to love mathematics. They are called recreational-mathematicians (like Dattaraya R. Kaprekar, Tanya Khovanova,…), maths-popularizer (like Eric Temple Bell, Constance Reid,
Simon Singh,… ), maths-historians (like Bartel Leendert van der Waerden, Jacqueline Anne Stedall,… ) etc.  But I want to become a mainstream-mathematician (like full time professors in research organizations). Why?

I want to become immortal (i.e. to exist as long as humans exist).

You may be think that I have lost my mind, but please continue reading…

I hope we agree that our wish to live (as opposed to fear of death) motivates us to live. Paul Erdős used to tell following story about his second discovery as a child (first being that of negative numbers):

I knew I would die. From then on, I’ve always wanted to be younger. In 1970, I preached in Los Angeles on ‘my first two and a half billion years in mathematics.’ When I was a child, the Earth was said to be two billion years old. Now scientists say it’s four and a half billion. So that makes me two and a half billion….

I believe that Paul Erdős has indeed gained immortality, we just keep listing about his conjecture being proved now and then (recent one: Erdős-Rado sunflower problem).

From a biologist’s point of view (by the way, I also study a bit of Biology), our body along with our consciousness defines us as an individual. So, for a human to become  immortal his/her body as a whole must be preserved as it is. But, biologists have faced a dilemma of  conserving body versus consciousness. If you preserve body, by eliminating defects from our body at DNA level (since not every organ can be transplanted) , the you lose the distinctiveness in personalities (like clones) since all will be perfect and thus identical (causing threat to evolution). If you preserve consciousness, by transferring it to an artificial body (which will become reality with advent of quantum computing), then you lose your body (a major part of your personality).

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Consciousness transferred to a robotic body in the sci-fi movie Chappie (© 2015, Sony Pictures Entertainment)

But, I believe that the only way to become immortal is by publishing (articles, books, movies, songs,…) and propagating (lectures, discussions,…) your ideas among others. As we know that our body is immortal in a sense that all its atoms remain as such (law of conservation of mass, energy…), and form various molecules like the molecules of life. For example, say an animal dies in a forest (without human intervention). After few days the microorganisms inside & outside the body (which are much more than the number of cells of that animal) will start decaying the body and release different chemicals. These chemicals will attract different insects which will start consuming the body and finally scavengers will completely clean the flesh part. The bones will take longer time depending on environmental conditions. Now these insects and scavengers will be consumed by bigger animals (and eventually, may be, by humans). In this way the atoms from the dead animal will disperse among various life-forms but will never cease existence. In case of humans, we make this process to take longer time by doing various rituals. So, if I am able to propagate all of my ideas (which will also evolve over time), then I am immortal.

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Goddess Saraswati (symbol of knowledge in Hindu Mythology) used to popularize a mathematics conference in Belgium (url: http://www.mathconf.org/app-gvl-summer2016)

The  immortality “ideology” which I want to adopt is actually what various civilizations have done to make their gods/goddesses immortal! You make a story about a character in written form (which they called “sacred texts”) and propagate among others (which they called “religions”). After some time the stories become part of our life and characters of that story become immortal. I view the “library of Alexandria” as power house of  Egyptian civilization, since it was a major center of scholarship. Same is true for eminent people (the people about whom biographies are written and movies are made) in modern society.

In my opinion, only mainstream researchers like scientists, psychologists, economists, etc. have an opportunity to gain immortality. Whereas people like non-innovative-teachers, librarians, science-popularizers, non-research physicians, non-research engineers etc. ensure immortality of others, just like the craftsman reproducing work  of ancestors again and again thus helping to keep the work alive.  So, all professions are about “collecting knowledge” but what makes researchers stand apart from other professions is their ability to “create knowledge“. So all professions are important but in different prospective. For example, if you want to become powerful, become politician and so on….

I admit that my thoughts may be very childish and I in future I may change  my opinion…

Numbers and Logic

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I am a big fan of number theory. I find the answer to Hilbert’s Tenth Problem fascinating. I was introduced to this problem, a couple of years ago, via the documentary titled : “Julia Robinson and Hilbert’s Tenth Problem“, here is the trailer:

 

You can read more about it here. Also for the sake of completeness, let me state Hilbert’s Tenth Problem:

Does there exist an algorithm to determine whether a given Diophantine equation has a solution in rational integers?

In 1970, Yuri Matiyasevich completed the solution of this problem by using the concept of Turing Machine. This short video provides a nice overview about Turing Machines in general

The answer to Hilbert’s Tenth Problem problem is

No such algorithm exists.

This interplay of number theory and logic is really interesting, isn’t it? But I can’t discuss solution of Hilbert’s Tenth Problem here, since I have never read it. But there is nice overview at Wikipedia.

I will rather discuss a puzzle from Boris A. Kordemsky’s book which illustrates the idea of this interplay.

Ask a friend to pick a number from 1 through 1000. After asking him/her ten questions that can be answered yes or no, you tell him/her the number. What kind of question?

The key to the solution is that 2 to the tenth power is 1024 (that is, over 1000). With each question you knock out half the remaining numbers, and after ten questions only the thought number is left.

I welcome you to think of a number and write the corresponding yes/no questions as a comment below.