 Now let us quickly summarize what we did yesterday. So we have formulated what we call as learning outcomes for this series of lectures in the form of a few questions which we are going to answer. Now yesterday we discussed the answers to questions such as what is the motivation and goal of a course such as introduction to research. Why do we need a course like this? So we said it is important to facilitate the development of right attitudes, skills and habits in research scholars so that ultimately we are able to achieve the vision of excellence in research. Then we enumerated or listed out what are the skills, habits and attitudes required. So we said that a research scholar should have good thinking skills, good communication skills then he needs to develop the skills of doing experiments, stress and time management and also doing vast amounts of literature survey. As regards the habits, we said a research scholar should have reading habits, documentation habit, document whatever new ideas you come across. So I hope you are all taking notes during this course and also participate in meetings and discussions. As far as the topic of attitudes is concerned it is a very important and much wider topic. We touched upon one important aspect namely that of motivation. If you want to do high quality research it is not sufficient if you are motivated by desire for promotion and things like that. So those kind of motivations are not sufficient for you to do great research. So you must be motivated by higher motives like intellectual challenge, intellectual satisfaction or service to society and so on. So we listed out those various motives. Then we went on to point out what is the difference between a course based education and research education. So we set out the goals of bachelors, masters and PhD and we said that if you do a PhD you take a license or you get a license to teach and guide others. Now we explained what does what ability should a person have for teaching and guiding others. Specifically we pointed out that you must have the ability to manage your own learning and be an independent thinker. Next we started on a discussion of good thinking skills we called it as productive thinking. So if you want to be good researchers we must become good thinkers and then we must understand what is meant by good thinking. So we said as against learning a large number of different subjects you should be able to, your mind should be able to conceive different ways of doing the same thing. So productive thinking involves coming up with a variety of approaches for doing the same thing. If you are able to conceive different approaches of doing the same thing then we can say that your mind will tend to be productive or it would tend to be creative. So now we will proceed further on this and discuss what are the different ways in which one can think. So that will be the discussion, topic of discussion for this particular session. So I am repeating the quote that we made yesterday that is the theme of this particular session. Education is not about learning diverse subjects but about learning diverse ways to the same subject. Now we shall begin with an activity meant to illustrate various levels of thinking. Now if one wants to be a good athlete then one must have some knowledge of how the body, the muscles and so on work. While one may not be an expert, a medical practitioner, you need not be a medical practitioner to be an athlete but you must have a sufficient knowledge of how your body works. Similarly, if you want to be a good researcher then you must know what are the various ways in which you can think, what are the various levels in which you can think. So you may not be a researcher on thinking process itself but you must have a knowledge of these things. So this is what we are discussing here. Now I will like to introduce this topic of levels of thinking with the help of an activity. So let me first state what are the levels of thinking that we want to understand with the help of the activity that we will carry out. So the lowest level of thinking is memory. Next higher level is called application or problem solving. The next higher level is evaluation or critical thinking. Finally, you have creativity. So what are these various levels? How do we identify whether the particular thinking that is happening is merely memory based or is it critical thinking or is it application or problem solving? So this is what we want to understand. Now the activity that we will do is something that looks very simple but as you will find it is interesting. Like five squares of equal size out of a single large square you are allowed to cut and paste. So what I am saying is that imagine that you have let us say a square piece of paper. All of you have a piece of paper you can probably create a square piece out of the sheets that you have by doing something like this. So here I have made a square piece of paper. Now let us say you have this piece and you are given a scissor. Now these are the only two things you have. You have to divide this square piece of paper into five equal parts. Please pay due attention to all the words that are used in this problem statement. So we are saying that you must divide this square piece of paper into five, number five. Five parts and all the parts should be equal in area and all the five parts should be squares. So divide the square piece of paper into five equal squares. So for instance if I want to divide it into four equal squares you know it is very simple. I divide this into half and then I further divide this half and then I get a quarter. Now if I tear out I will get four equal squares, one square, second square, third square, third square here and fourth square. Now the puzzle is to divide into five equal parts. You are allowed to cut and paste. What does it mean? If you feel that you have a method in which you divide it into maybe hundred parts and then you will join together twenty parts make one square, another twenty parts make another square and so on you are permitted. Please divide into as many parts as you want but ultimately you can paste them together and then make five squares. So that is the puzzle. Let me draw it out and further explain this point what you are expected to do. I am not giving the solution. I am just trying to clarify what I want you to do because whenever I have done this activity I have found that people miss out on the nuances of the problem statement. So let me draw it out. So this is my square piece of paper and ultimately I want this kind of a thing. One square, second square, third square, fourth square and fifth square. So one, two, three, four and five. So if I sum the area of this five squares I must get this big square. That is how it should be. So it is a fairly straightforward statement and what I have said is that you can divide it into more parts if you want that each of these squares may be constructed from different parts. It may have different pieces and then you are joining them together. If you want it that way you are permitted. So before proceeding on the activity I will just take a couple of questions on the problem statement. In case you do not, you have not understood the problem statement please seek a clarification. So just two centers just to see whether you have understood the thing correctly. Let me, so let me go to Walchand Institute, Walchand Institute, does someone want to give a solution? Let us consider my square area for example 10 into 10 hundred. This is my original square. So I will divide this into five equal squares by putting equation like this. Let us consider x is the side of my first square. So the equation forms like this x square into 5 equal to 100. So 100 unit of original square. So after solving this equation I will get side x is equal to 2 into square root of 5. So this will be the side of my first square. So I hope the addition of all 5 squares results the equal area of original. Whatever you have said is not really the solution because I have told you you have a scissor and you should actually realize that 2 root 5 that is the important point. How will you cut it so that you get that length of 2 root 5? You have to actually cut and show me that you have got the 5 squares. You understand? How will you get the 2 root 5? That is the big problem. I hope I will get in this way sir, I have not tried but maybe this is the x. So that is 2 into square root of 5. So this will be the one part similarly right side top, second part and third part left side below and the fourth part right side below. So 4 equal square and the remaining part area by addition of all 4 pieces will result fifth area. This may be. Now let me just before moving on to another center let me just repeat what you have said and tell you what is the problem with that. What you are saying is that you will divide this is your square. You will have one square cut at this end, another square cut here, another one cut here and cut here and the remaining part you say somehow you will make the fifth square. Now this is really not a solution because first of all I do not know according to you this one side is 2 root 5. I do not know how you are going to get this 2 root 5 that is one problem. Second problem is it is not clear how this shape that you get in the middle can be converted into a square by cutting that is not clear. So please understand that when you say you have a solution you should really tell me how the whole thing is being cut and you should show me the 5 squares right and if you cannot get the solution at least you should tell what is the direction in which you are moving like for example you have told in this case right. It is important for us to see how you are thinking about this problem. Solution is not important how you are thinking and you recognizing whether this approach will lead to the solution is important. So here I do not see how it will lead to a solution because I do not know how the shaded area can be converted into a square by cutting whichever way you want to cut. Let please retry I will go to another center. So let me go to Vignan Institute of Technology and Science, Nalgonda. Hello. So first divide the sheet into the four parts like this. This is a four parts then after this each part it will be divided into the five parts then you can cut the four parts this it will be another part this it will be the exactly because this is exactly equal area. So please repeat what you have done you have. So I it is they folded into the four parts and each part it is I divided as the five parts and it is at each edge I as a marked as the four four and it is a cut down in the middle you will having the on the top two parts and the bottom two parts if you can combine it is a one square. Now I got the point that you divide as paper into four parts please take one fourth of the paper and please repeat the remaining thing I did not get how you divide the each quarter of the paper. The fourth of the paper is a divided into the five parts. Yeah show me how you divide into five parts. Five rectangles. Ah five rectangles okay five rectangles then. Five rectangles okay five rectangles suppose if it is a this is like that it is a five rectangle side okay when you cut each side I am cutting into the each side it is a four rectangles each side it is a four rectangle this side it is a four rectangle you will having the two rectangles here two rectangles here if you can combine this two to four rectangles you will get the one square. This is the you can measure it is exactly it is the. Yes please let me get that part is not clear you are saying you are dividing let me draw it out and then you tell me whether I understood it right there is a little bit of problem in understanding what you exactly you have done. So you are saying that first you take this square you divide into four parts now you take this small part one fourth of it you divide into five rectangles now that is not or do you divide into five rectangles like this or which is the other way you are dividing that is not clear am I right okay vertically yeah you are saying like this is that yes okay okay then. So that you can count it is the four rectangles in each side in this side it is a four rectangles and it is the down also four rectangles and it is the four rectangles. So that is exactly you will be there two rectangles in the up. Let us see can you is there some way they can they can draw it and show me we must learn to interact in this mode so it is worth spending little bit of time can you draw your solution on a piece of paper show me how you divide the rectangle just draw lines and then show over the you know project it to the let the camera capture what you have done yes good good now I am able to see yes okay. So this is the four corners you will guard the four squares correct and it is the it is the in middle one you can combine you will get one more square. It is not clear how look how do you get a square out of the central rectangle it is having the eight two rectangles and each side rectangle it is having to the four rectangles. So I think there is a so I divided the first to the paper and are you dividing the quarter into rectangles or squares because sometimes you are using rectangles as the word to describe what you are doing and sometimes you are using squares please mark on the piece of paper squares with an s if it is a square you mark an s there and tell me that is an s square this is also an exercise for you in communication how do you communicate quickly what you want to say hello yes this is the all rta it is a you will having the squares only okay maybe I will just go to another center right because now I have got the gist of what you are saying you are dividing the paper vertically and horizontally please try it out I think there will be a problem you will not get the five squares I am just giving you a hint let me just go to another place Pune D. Y. Patil now I am going to D. Y. Patil Institute of Engineering and Technology Pune my question to you is a center in Andhra Pradesh just now described a solution okay can someone tell me whether they could clearly see what the person was talking about yes sir the square which is given of dimension x the area of the square is x square so we are dividing it into the phi u equal square so the dimensions are x square by phi u that is area of each square so each square is of dimension x by root phi u so we can cut corner of size x by root phi u and remaining area is of x square by phi u which we can join and form the one fifth square. Now see this is a repetition of the solution given by an earlier student the problem here is how do you get x by root 5 problem number one I have told you you have a scissor and you have a paper you have nothing else how do you measure x by root 5 problem number 2 the central portion is not a square you are saying that it is its area is equal to that of a square but how do you convert that into a square shape right I want to see 5 squares you understand it is not going to give me 5 squares the way you are saying first of all I do not know how to get the length x by root 5 okay let me go to another center we actually are running out of time but still I will take one more center we need to go further and discuss now I would like to you to tell me quickly if you are following a scheme where you are not dividing the paper vertically and horizontally right because one method of dividing the paper is vertical and horizontal all of you seem to be following a vertical or horizontal division parallel to the edges okay now I will go to a center and that center should tell me if it is following a different scheme otherwise we will not take the solution we will go to another center right okay let us go to New Delhi yes Bhagawan Parishoram instead of technology New Delhi yeah shall it be possible solution of this because we have tried many but we are not getting okay fine yeah there is a solution definitely otherwise I would not have given you the problem okay it is but it is not a trivial puzzle okay you may take time to do it what I am interested in is the activity okay I think what we will do is we will proceed with the remaining part of my our discussion on productive thinking because actually solution of the puzzle is only a part of the whole thing right it is not necessary to get the solution to proceed further now I said that we would like to explain the various levels of thinking using this activity that we have carried out so let me go to the next slide and if you have been trying this solution for this puzzle please stop because we want to discuss further and you can resume the your you know trial later on because I am not giving a solution so that you have the pleasure of solving the puzzle okay now though I am not going to give you the solution I want to tell you that one important hint for solving the puzzle is that you must apply Pythagoras theorem okay to get the dimension such as root 5 or 1 by root 5 and so on now you know how Pythagoras theorem allows you to get a dimension such as root 5 we might think that root 5 is an irrational number it is not possible for you to geometrically achieve this kind of a length but this is not correct because you know that if I take a right angle triangle in which one side is of length 2 another side is of length 1 then the hypotenuse will be square root of 1 square plus 2 square that is square root of 5 so in this manner you can realize a length of root 5 now more than this I will not tell you because I do not want to rob the pleasure your pleasure of solving the puzzle but now with this background we can explain the various levels of thinking so in summary we can solve this puzzle by application of Pythagoras theorem okay now supposing we have taught students the Pythagoras theorem and some proof of Pythagoras theorem something that we commonly do in all school curriculum now the important thing is what kind of a question do we set to test these students thinking so one possible question that one may set is so we are going to frame different types of questions on Pythagoras theorem and explain how the different questions test different levels of thinking so if you set a question such as state and proof Pythagoras theorem right so we do that in an exam we teach students Pythagoras theorem and teach the students proof of Pythagoras theorem and then in the examination we ask them state and proof Pythagoras theorem now this kind of a question only tests memory because we have already told given all the information to the student the student has to recall this information during examination so really this is not a problem though we may say that you know we have set a problem for the students but strictly speaking by the definition of the word problem this is not a problem right it is because problem is something that is not familiar to the person so state and proof Pythagoras theorem is a question that tests memory now supposing we set a different question that is shown on the slide the next one the second question using Pythagoras theorem divide a square piece of paper into five equal parts five equal squares now this is a more difficult question to answer it tests application or problem solving so this level of thinking is called problem solving or application where we are telling the starting point that is the Pythagoras theorem and we are presenting an unfamiliar situation so we assume that we have the students do not students have not faced this problem earlier division of a square piece of paper into five equal squares so then we say you start from a Pythagoras theorem and see how you can apply this theorem and divide a square piece of paper into five equal squares so this is a difficult question it is testing application or problem solving ability you can make the question more difficult like I did for you do not tell that you have to use Pythagoras theorem so teach students Pythagoras theorem and proof and then ask them to solve this answer this question divide a square piece of paper into five equal squares so no hint of Pythagoras theorem now this is a more difficult question to answer now this question tests what we call as critical thinking because no starting point is provided typically this kind of questions you know now you have thought over it for 10 15 minutes so what you are attempting is different approaches to the problem right so you consider one method of dividing the paper you divide horizontally and vertically and then you realize that oh no you are not going to get five squares out of this kind of an approach then maybe you try different scheme of division divide into triangles for instance can you divide into triangles or any other shapes and then combine triangles to get a square so you will try a different scheme so in this manner you will be trying different approaches and then you will try an approach then you will evaluate whether this approach is going to lead you to the solution realize that no it does not lead you to solution then discard the approach and try a new approach so this thinking in which issue is approach from different angles and each angle or each approach is evaluated for efficacy that is you decide whether this approach is going to lead you to solution and then you discard if it is not leading you to solution think of a different approach so this is a higher level of thinking than application of problem solving this is called critical thinking and even higher level of thinking is involved in setting a question to test various levels of thinking so for example now we have set a question like divide a square piece of paper into five equals squares and we are trying to see what level of thinking the students have so you give them some time if they are not able to solve this problem give them a hint say that you know you use Pythagoras theorem and try to solve it then see how many students are able to solve and so on so formulating or devising a question is a much higher level of thinking creativity so setting good questions is not easy it requires highest level of thinking for instance setting questions for an examination like the joint entrance examination to IITs requires a very high level of thinking so it test the creativity in people so with the help of this activity and these set of different questions that are shown in the slide I hope you will gain an appreciation of different levels of thinking how depending on how the question is framed you can test various levels of thinking in individual including ourselves we can test whether we whether most of our thinking is memory based or are we able to apply concepts to unfamiliar situations that would be application of problem solving do you have that ability do you have critical thinking ability if you are not given a starting point at all and you are simply presented with the problem right in research this is the kind of situation that we encounter so we say that in research we do higher order thinking namely critical thinking and creativity even higher than problem solving okay so here is an assignment for you understand critical thinking in detail using an internet search because research involves lot of critical thinking and of course creative thinking also creative thinking we are going to discuss in our remaining part of this sessions on productive thinking but critical thinking you try to understand yourself by doing an internet search so you put Google search critical thinking you will get lot of material okay so please do this exercise let me introduce this topic of creativity what is creative thinking okay we have said during the beginning of this session on productive thinking that it is by asking new questions that you make significant contributions okay so creativity requires you to ask questions that is one of the ways in which creativity is comes out right if you ask you are able to ask good questions then you can be creative but there are many other things that are required for creativity so let us define what is creativity okay so we have for example defined what is meant by productive thinking coming up with different approaches for doing is the same thing we are broadly talking about the same higher level thinking abilities in terms of different words that are commonly used in practice a creative thinking is another word that is used creativity creative thinking and so on so let us define what this thinking means it is ability to look at the same thing as everyone else and think something different this is one definition given by psychologist for creativity another definition given is a the second statement there on the slide ability to take a fresh look at familiar objects and situations which is enriched by past experience but not constrained by it okay please spend some time in thinking about this definitions let us take the first one ability to look at the same thing as everyone else and think something different in other words thinking differently is a mark of creativity illustrate this with some examples right let us take this example thinking differently here is an example of thinking differently ability to look at the same thing as everyone else but think something different now all of us have seen roses and thorns now someone who also saw roses and thorns like us came up with this idea so here we have listed on this first column various attitudes that human beings possess like optimistic pessimistic realistic stoic and so on and on the right hand side we have some sort of definition of these attitude using only roses and thorns only two words roses and thorns so optimistic is equivalent to seeing roses everywhere right so optimistic is roses pessimistic you see thorns everywhere then you say it is a pessimistic attitude towards like supposing you see the world as full of roses and thorns you know sometimes you have roses sometimes you have thorns then you say you are being realistic stoic how does it matter whether there are roses or thorns in life it does not matter right if you have that sort of an attitude you say I do not care whether I get roses or I get thorns I am always steadfast right stoic human roses for you and roses for me so let there let good things happen to everyone then you say you are human selfish roses for me and thorns for you so all the good things should come to me and all the bad things should happen with you so that is selfish sadistic thorns for you and your blood for me you enjoy if others are in trouble that is a sadistic attitude and finally divine roses for you and your thorns for me that is how a saint that is a saintly attitude so where the saint says that let all good things happen to you and I will bear all your difficulties so you see using two words roses and thorns someone has conceived definitions of all the various attitudes that we encounter in life and distinguish between them okay now this is a creative piece looking at the same thing as everyone else but thinking something different so this is how in fact we can look into ourselves and see whether we have this ability have we ever thought differently than others if we have then only we can say we are creative similarly if we are teachers we should try to see in our own students whether they have this ability to think differently so as you can see this kind of abilities are not tested very easily in examinations okay to test whether people have creativity and so on it is not so easy but it is not that we can we need to assess people only through examinations and particularly the examination system that we have okay there are many other ways of assessing creativity in individuals in interactions here are some more examples okay same line of definition ability to look at the same thing as everyone else but think something different and now this is the first thing shown in the slide is about naming of individuals let me tell you a small story regarding this now in our institution some years ago there was a student whose name used to come up very often in all our faculty meetings particularly those meetings which were supposed which were held for finalizing the grades the reason was this student would fail in most of the courses the name of the student was Ma Shivayana Ma Shivayana now this name was very it was unique name we have not come across such a name so once a discussion ensued among the faculty you know what does this name mean then a faculty member pointed out that he had in fact discussed this point with the student and asked him you know how were you named and then this is what he said what he said was very interesting so he said his father is a poor farmer and mostly illiterate but he is an ardent devotee of Shiva and he somehow had a dream he had a dream that he is going to get five sons and he wanted to name all the five sons after Shiva so he came up with this scheme he said that if you take the word Na Ma Shivaya if you write it in an Indian an Indian script such as Hindi for instance so it would be Na Ma Shivaya right so there will be five letters in the Indian script so this is what is written in the roman script here in the slide so Na Ma Shivaya so the first name he decided he would name Na Ma Shivaya now the next son he said I will rotate I will start with Ma instead of Na and then it will be cyclic right so next son would be Ma Shivaya Na then the third son would be Shivaya Na Ma the fourth son would be Vaaya Na Ma Shivaya and the fifth one is Na Ma Shivaya and he said so actually he has four brothers and they are named like this so now this is very interesting right it is an example of creativity because you have not we do not see people coming across naming we do not come across people naming their children in this fashion right so here is an example of creativity and some more couple of examples are shown here I have deliberately chosen examples which people of all disciplines will understand one can talk of great creative works in science and engineering which is what you should do but we should understand what is the definition of creativity from which angle we have to look at the work done by individuals to appreciate where is the creativity involved it is in the in the different method of thinking that is where creativity lies so here are some more examples creativity in expression the ambassador from United States to India when he was leaving one of the ambassadors I forget the name during his speech he was trying to express his warm for the Indian people so he said it as follows so he said when I go back to US during the coming winter our vivid memories of India will warm us as we face the snows so if you take word by word I am sure all of us know each word and you know we must have used it in some sentence or the other but coming up with a composition like this to express your warmth putting these words together in this fashion is where the creativity lies now finally one more example of humor you know humor there is lot of creativity in humor so there was someone who did research on capitalism and communism some social scientist and then he said what is the conclusion of his research he found that according to him his research the ultimate effect of capitalism or communism on human beings was the same he found there is nothing much there is one each one has advantage and disadvantages but some total of it is that both are not really good for people for human beings so he framed his conclusion as follows he said capitalism is exploitation of man by man and communism is exactly its opposite so it is also exploitation of man by man opposite of this is also exploitation of man by man so this is a very nice way of expressing your conclusion right so in expressions in language and so on also you find creativity so thinking differently is what is the mark of creativity now let us look at things as engineers and scientists little bit of technical matter right so we said that creativity is thinking differently we said that productive thinking is essentially thinking in different ways about the same issue or problem so let us to consider some examples of how we can think in different ways so some general thinking strategies are given here with examples reformulation or restatement so you have a problem you want to solve you can go on changing its statement right in order to solve it so you may find that you frame the problem in a particular manner its solution is not so easy you change the statement and then you get a solution quickly in fact very commonly in research we go on formulating and reformulating our research problem right so that we are able to provide an effective solution I am going to discuss an example of each type of thinking strategy in subsequent slides first let me state the strategies here graphical and tabular representation so many times you want to use some information or data to arrive at a solution how you represent the data is very important in getting the solution representing the data in tabular form or graphical form is a powerful method of coming up with solutions logical reasoning is another strategy commonly all mathematicians use this in proving theorems division into sub-problems so you have a big problem to solve you first divided into sub-problems and then solve each small problem and then you get the solution of the bigger problem by combining the solutions and finally using analogies to solve problems so we are going to discuss these strategies let us take one problem so I am going to state problem and then discuss a solution there could be other ways of solving the problems right so I will leave them to you as an exercises so let me tell you a brief background to this problem in the in Japan a particular well-known company I am forgetting the name of the company once found that it was losing out to competitors in terms of productivity so the management called a meeting of all the workers and it wanted to discuss how the productivity of the company can be improved so they had a plan they felt that they could ask the workers to give suggestions and to motivate the workers they said well what you do is you write your suggestions for improving the productivity on a piece of paper and drop it in a box okay and then the management will look at all these suggestions and if a particular suggestion is found to be promising it would try to implement the suggestion and if really the productivity went up then a part of the profits that the company makes will be given to the individual who has made the suggestion so the management thought that this should be sufficient motivation for the workers to come up with ideas and the company thought that it must involve the workers because they are the ones who are actually doing the work doing the production so they thought good suggestions would come from them now surprisingly what happened was that first time they called a meeting they got very few suggestions from the workers about improving the productivity now they were taken by surprise so someone in the management did a silent investigation into this and then he found that the reason workers were not responding was because they thought that it was a management trick to extract more work out of them so they felt that you know being to become more productive they will have to work harder right and they didn't want to do that why should they work hard for somebody for the company to make profits unless they get a share of share of it so they thought it is a management trick to extract more work out of the workers so if you involve the workers themselves into making suggestions then the workers would be probably more amenable to doing harder work so it was because of this fact of psychology that the workers did not respond the manager someone in the management realized that the problem lay in the framing of the question so the question that framed was how can you become more productive and this framing of the question was not really a friendly framing so then this person altered the statement of the question and after about six months they called another meeting now this time they asked for suggestions on the following question so how can you make your job easier so they said they asked the workers you know how can you make your job easier we want to know now if you carefully see both these questions are actually equivalent in some sense because if a particular job becomes easy for the workers then in the given time the workers can do more jobs so evidently that would improve the productivity now as it happened the workers gave a lot of suggestions so this time the workers felt that you know this was a question for which they can give suggestions and it would be their to their advantage so lot of suggestions were given and the company actually implemented the suggestions and it is said that its productivity went up so whatever I described to you is a real thing that had happened although I do not remember the name of the company so you can see how framing of the question is very very important framing of the problem how you frame the problem is important one particular method of framing may not lead to solution whereas a slight change in the statement can make the solution easy now there are changing the statement casting the information in different forms is actually the basis of many powerful transform techniques that are used in mathematics for example binary number system logarithm these are examples so we want to multiply two numbers you convert the numbers into their logarithms and add their numbers and then you take an antilog that is what people would do maybe 30 40 years ago nowadays of course you have calculators but essentially to make the point so logarithm changes a multiplication operation into an addition operation okay so you take numbers you take their logarithms so logarithms are the same numbers in a different form okay similarly binary number system you have decimal number system you can express the same numbers in a different form namely binary form and we know that all computers and so on use a binary system right so this is how by transforming the information in a different form by changing the statement of the problem to a different form you may be able to get solutions easily so this is a very powerful method of getting solutions now here is another problem right and let us see what way we can solve it best let me first tell you the statement of the problem exactly at sunrise one morning you set out to climb the Tirupati temple the path bound round the mountain you climb the path at varying rates of speed you stopped many times along the way to rest and to eat the fruit you carried with you you reach the temple just before sunset so you started in the morning went up the mountain in some roundabout manner and then you reached the temple at the sunset after fasting and meditating for several days you began your journey down along the same binding path starting at sunrise and walking as before at variable speeds you your average speed down the hill was more than your average climbing speed right this is what normally happens we always are able to walk down faster now what you have to do is prove that there must be a spot along the path that you will pass on both the trips at exactly the same time of the day so you are starting in the morning and reaching the top of the temple going round some path which may not be straight then you are starting from the top and reaching the bottom you are starting the top in the morning and then reaching the bottom in the evening or may be much earlier because your average speed down is much more than the average climbing speed now what you have to prove is that you will be at some spot at the same time of the day when climbing up and climbing down so please think over it how best you can solve this problem okay I will leave it as an exercise the strategy is available for you look at this slide that I put here you change the statement of the problem that could be one strategy to get a solution or use a graph to represent the information in this case may be a table representation may not be possible or you may use logical reasoning okay so let us discuss graphical representation logical reasoning in detail but for as far as this problem is concerned I will leave it as an exercise to you to use try this techniques and solving okay now this is the problem that I will discuss the solution for let us say divide derive the trend in the behavior of plating adhesion on a silicon substrate from the measured data as a function of substrate area and doping level now this problem has been framed in the context of semiconductor substrates so there are some words here which may be specific which may not be understandable by people in other areas but that really does not matter I will draw a diagram to illustrate what we are talking about basically you have a substrate and you are plating the substrate okay and you want to measure the adhesion of the plating on the substrate this adhesion will be a function of area and function of substrate conditions so from this data collected you want to check whether the adhesion varies in any particular pattern as a function of area and substrate condition so that is what this problem is all about so the adhesion is measured for areas of point five one and two centimeter square and p plus p and n plus doping levels so if you do not understand doping you it does not really matter so it is talking about let us say the conductivity of the substrate so p plus substrate is a substrate of high conductivity in which the current is carried only by positive charges p type substrate is a substrate in which current is carried by positive charges but the conductivity is not so high similarly n and n plus substrates mean these are substrates where the current is carried by negative charges like electrons and n means a moderate concentration of charges whereas n plus means a very high concentration or very low conductivity sorry very high concentration very high conductivity so basically you are varying the conductivity of the substrate you are also varying the polarity of the carry charges which are carrying the current and you want to know whether these have any effect on the adhesion now you want to derive the strength by actually measuring the adhesion for various substrate conditions and various areas and from this you want to derive the pattern and each measurement you want to repeat at least twice or twice you know every measurement you do a number of times to remove the random error in the experiment in the measurement so how will you go about representing the data the measurements so that you get the pattern out of it so that is what we want to see now we are close to the break I will probably take up this the solution to this problem after the break