 started on the skills associated with thinking and in thinking we have seen various ways of solving problems that is what we are discussing now. So far we have discussed the representation skills in some detail. So for example how you can use symbols instead of long statements for describing situations then we have seen how you can use a table to represent data then how you can use a graph to represent data and how a graphical representation is very powerful in giving you a lot of information. We have also seen how you can restate a problem so as to solve it easily. Restate or reformulate. Now proceeding in the same direction let us see a few more methods of representation. A few more things that we need to look for when we represent problems. Let us begin with the Venn diagrams. All of us we have studied about Venn diagrams in the school but surprisingly people do not use Venn diagrams to represent relations between various entities which is what it is supposed to do when they are actually embarking on solving problems for research or even for understanding various things. So here are two examples shown on the slide. Supposing we want to show the relation between the activities of science and inventing, scientists and inventors. So here one of the circles here shows the group of inventors and the other circle is a group of scientists. Now when you show overlapping circles what it means is there are some people who are scientists but not inventors. That is this group. You can have people who are scientists as well as inventors because science and invention are not the same activities. So there is some difference between them but that doesn't mean that the group of scientists and inventors are totally distinct. So you can have people who are doing mostly inventing but not doing activities of science as we have understood. There can be some group of people who are doing science and invention and there are some people who are doing mostly science not much of invention. Here is another example. In physics or even in engineering some courses which are very close to physics we use two terms called equilibrium and steady state. I have often seen people getting confused between these two words equilibrium and steady state and therefore they use these two words interchangeably. If there are any students who are from the area of solid state devices or from the area of physics they will appreciate this much better. So what is the meaning of equilibrium state? Equilibrium state means it is a state in which for every process there is an inverse process going on at the same rate. That is the definition of the equilibrium state. For every process there is an inverse process going on at the same rate. What is the definition of steady state? Steady state means all processes are constant as a function of time. So steady state means all processes are constant as a function of time. Now you ask students a question. If a body is in equilibrium is it in steady state? If a body is in steady state then does it imply that it will be in equilibrium? And many times when these kind of questions are put the students are confused. So this is a Venn diagram which represents the relation between steady state and equilibrium. So here equilibrium is a subset of steady state. This means if a body is in equilibrium it has to be in steady state. However if a body is in steady state it may or may not be in equilibrium because all processes may be constant with time but it may not happen that for every process there is an inverse process going on at the same rate. So like this one can use Venn diagrams to represent relations between various quantities particularly those which appear to be appear to have some commonality. In fact there are more complicated Venn diagrams because in our course we want to spend some time on each of the various methods of representation. I will not discuss this topic of Venn diagrams in more detail. But you can whenever you have difficulties in distinguishing between two different entities and so on you can try to see if you can represent things on a Venn diagram. Venn diagrams can be used for very many situations. Let us see another aspect that we should consider while representing problems. Drawing diagrams to scale. We have emphasized this the importance of drawing diagrams to scale yesterday when we were discussing solution to the puzzle of dividing squares into five equal parts. How if you do not draw diagrams to scale you may not be able to see the fact that solution is not possible. On the other hand other thing is also possible that is when you draw diagrams to scale you may see a particular approach of solving a problem which you will not be able to see if you do not draw diagrams to scale. So here is an example. Now supposing we want to model the current flow between two contacts. Okay a small contact at the top and a larger contact at the bottom. Now what are shown here are current flow lines between the two contacts. Do you have to take into account two dimensional effects? Suppose this is the question you have. When you want to model the current flow between the two contacts do you need to take into account two dimensional effects? Because if you have to consider two dimensional effects evidently the model will be more complex. Now it all depends on the scale. So here two situations are shown. The distance between the contacts and the size of the smaller contact. These are the important things components of this particular problem. Now this is case one supposing the relative sizes of the length of the contact and the distance is as shown here 8 and 11. Now when you draw the flow diagram you can clearly see that you will have a significant amount of current which shows which flows along those curved lines where two dimensional effects are there. However if your situation is as shown in the second diagram here below. Where the distance between the contacts is less than even one tenth of the length of the contact. So now you see that the extent to which the two dimensional flow or the two dimensional region extends beyond this one dimensional region is very very small. So in this case in fact if you want to model the situation you can simply use a one dimensional approximation. Normally what happens is we do not draw diagrams to scale. So even though our contact size may be 8 and the distance may be 0.11 as shown here you will draw a diagram something as shown in the first. Okay and you'll simply mark the values as 8 and 0.11. So first you draw the diagram and then you simply mark the values right when you even when you're explaining without taking care to draw the diagram to scale. So as a result you will miss such important points. In modeling in fact it is very often seen drawing diagrams to scale helps immensely in deriving simplified models or in making proper assumptions. Okay so that your model will be simplified. So in fact I can give you one more example from my own modeling experience. In a particular situation there was a device in which there were charges located at various places. So for example there was a positive charge here then a negative charge at the other end then positive charge at the other side of the same surface for some other reason and a consequence of these positive and negative charges was an accumulation of electrons here. The model involved calculating the number of electrons as a consequence of these charges. Now the device used to be drawn as I have drawn it here. Okay so where the relative distances were actually not drawn to scale. In fact you see a diagram you very rarely find diagrams drawn to scale. Why? Because the relative distance was if this was one unit this was thousand units. Now evidently if you try to draw a diagram to scale you will not be able to show this presence of these various charges in the device at all. However for the particular problem at hand which involved finding out the effect of these charges on the electron concentration actually the details of the individual charges here how many positive charges are located at this point how many negative charges here and how many positive charges here. These individual details are actually not important. This is clearly seen if I draw the same diagram to scale. If I want to show this is just one unit this entire thing is just this one unit right and the remaining portion is thousand units length. Now what it means is all these three charges are in this compressed into this thin region and then you have the electrons created as a result of this. So what matters is not the individual charges but the sum total of all these charges. So that makes this problem very simple because you don't have to model individual charges separately and the phenomena associated with them. All that you try to see is what is the sum total of all this. So like this one can give many examples where if you draw diagrams to scale you can derive models or you can solve the problem in general in a general case in a much more effective manner. Completed the discussion on representation various ways in which you can represent problems and what are the aspects that we should take care while making a representation. Okay now let us move on to the other strategies of solving problems. Let us consider this particular problem. How many matches should be played on a knockout basis to decide the winner from among ten teams? Suppose this is a problem. Now one way you could solve this problem is to draw a graph. So here we have shown the ten teams first and then you show branches of a tree joining together so where a node means a match. So now you have five matches at the end of which five teams are left. Then you arrange matches between every pair of teams. One team is going to be given a buy because you know the number is odd. So you draw a tree like this and then every node you mark it as a match and then you count the number of nodes. Okay so you find from here that you need to play nine matches. Actually there is a simpler method of solving this problem. If you use logical thinking to solve this problem what is logical thinking or reasoning? Example of that is this statement. All men are mortal. Rama is a man so Rama is a mortal. So first you have a proposition then you have another statement and then you relate the two and derive the third. Let us exactly use this reasoning in the present case. First statement. To decide a winner from among ten teams, nine teams have to be eliminated. Only then you are left with a winner. Every match eliminates one team. Hence nine matches need to be played for eliminating nine teams. So this is a solving of the problem by logical reasoning. Now what is the advantage of this approach as compared to graphical approach? You see logical reasoning it is extendable. Supposing I was given N teams and you are asked to derive a formula. Right how many matches need to be played by a knockout basis. So to decide a winner from among N teams, N minus one teams have to be eliminated. Every match eliminates one team. Hence N minus one matches need to be played for eliminating N minus one teams. So you see you get a general formula that if there are N teams you should play N minus one matches. Something that is not very evident when you draw a tree. Right that if you count all the number of nodes it will they the number of nodes will turn out to be simply N minus one. This is not very evident if you use a graphical approach. So logical reasoning is also a very powerful method of solving problems. And we do not use it as frequently. Give you one more example. So the problem statement is as follows. The human figures painted by El Greco seem unnaturally tall and thin. El Greco is some famous painter. Now this is an observation. So an ophthalmologist one who deals with defects of the eye and how the eye works and so on. He surmised that this is because El Greco suffered a defect of vision that made him see people that way. Because he was trying to explain why should El Greco paint human figures very tall and thin, unnatural. Now is this argument correct? Right. So this argument is not correct. So how do you derive that argument is not correct? Because El Greco has a certain way of looking at things that suppose they are elongated for example. Now if he draws a proper figure it will look elongated to him which will appear normal. So for it to appear normal to him it has for him for it to appear normal it has to still be normal there. The only thing that differs is his perception of normalcy. Can you just repeat? Yeah. Can you repeat what you have said? For it to appear normal to him it has to be elongated. Or maybe compressed. I mean if the defect of vision was there. Let's assume that it is elongated. So for it to appear normal to him it has to appear elongated. If he draws it normally it will appear elongated to him. Therefore what he'll try to do is to make it look normal for himself. In which case he'll make it he'll make the drawing look normal so that it looks elongated to him which is normal for him. Right. So the point is it cannot be because of defect of vision. Okay. So this is you derive it just by logical reasoning. Another way of looking at the same thing would be right. If for example there was someone has a defect of vision that he sees things double. Now if you say that this defect of vision will prompt him to draw things double on the paint things double. But then by logical reasoning when he sees two things which will appear to him as four times. So what is going to happen? Right. As he paints two things they will appear to him as four figures. Right. So evidently therefore it is not possible that because of defect of vision a person will paint things double or elongated. Okay. So it has to be intentional. So it is not that his defect he had a defect of vision but for some reason intentionally he drew them like that. Okay. So many such examples are there. One can take very many sophisticated examples particularly proofs of theorems. Where you can show by sophisticated chain of reasoning which may involve only two or three steps. You can prove theorems. Okay. You can even understand things by logical reasoning. Let us consider one more strategy of solving problems. Okay. Many of these things are picked up from my own experience. So this is one problem. Right. Why is the computer down? I had an interesting experience. One of my colleagues from chemistry department and friend he called me and said Shri Pat can you just come and help me. My computer is not working. Okay. So but I said why do you think of me for finding out why it is down? No you are an electrical engineer. Maybe you will know why it is not working. Okay. So the idea here was that he is a chemistry person and if you want to find out why the computer is down what may be the problem. You need specialized knowledge of electrical engineering. Okay. At least that is the way he saw it. Now I have chosen this problem because I have had five such requests over last ten years. Right. By different colleagues and from other departments. That is very interesting. All of them calling me and saying can you just check. Help me. I know why the computer is down. Because they seem to think that it requires the knowledge of electrical engineering to check this. Now for example one situation. What was the problem? I actually went and you know said the problem right. But involved nothing of electrical engineering. So what does it involve basically? What will you do? You will draw a diagram and show all the elements. Okay. So this is the socket. Then you have the plug. Then you have this wire connecting the UPS uninterrupted power supply. Another wire running from this power supply to the computer. Now what will you do? The computer can be down because of many possible reasons. Right. So either this wire might have gone back or the UPS is not working or this wire has gone back or maybe the electricity is not there or maybe the computer has got spoiled. That is also possible. Now you will eliminate one by one the various possibilities. So first you will see that other lights are on. So obviously the electricity is there. Fine. Then you can remove this computer and connect another device over here and see whether that is working. Electrical device. If that device is working then you conclude that maybe the computer is back. But if the device is not working that means there is some problem in this part. So you go on tracing. Is the wire good? Is the UPS all right? And so on. And then in one case, for example, I found the problem was that the wire inside the plug had come out from one of the pins of the plug. Because you are constantly removing the plug and inserting it, connecting the computer, disconnecting it, moving it to a different place and so on. So all that you needed to do was to just fix that point. The idea I want to convey is that in many cases problems involve simple analysis. Now we use our analysis so often. I am sure my colleague would have taught analysis in chemistry and so on. Analysis of reactions and so on. But that does not necessarily mean we use analysis to solve problems. In very many situations it is simple analysis, logical reasoning, separation of the problem into smaller parts and understanding the parts in isolation. This is the meaning of analysis. What is the meaning of analysis? Separation of whole into parts. Understanding the parts in isolation. Then combining the understanding obtained for the isolated parts so as to understand the whole. That is what is the analysis. Now if someone asks us to explain what is the meaning of analysis also we may be able to do. But the moment we are faced with a problem which is not familiar, this is very important. The moment we are faced with a problem which we think at least in our perception is not familiar, we are not comfortable with, then we immediately think that specialized knowledge of some area is required to solve the problem. Whereas many problems are solved by general problem solving strategies. Logical reasoning, separation of the problem into parts and so on. So we must attempt these approaches before thinking that we should use specialized knowledge to solve a problem. Let us look at one more skill that one employs to solve problems. So I have to explain it but it is not very difficult to understand because all of us here I think know what is the meaning of a resistor. That is all what you need to know to understand this problem and its solution. So consider a Wheatstone bridge with three resistors of value r and one resistor of value r plus delta r and two formulae are given there which tries to relate input voltage and output voltage. So let us draw a diagram for this situation. So this is a Wheatstone bridge. So you have four resistances here. This is the input voltage and you take the output voltage between these two points. When this bridge is balanced so this is one arm of the bridge and this is another arm of the bridge. So when the two arms are balanced that is if all these four resistors are of value r then this voltage will be 0. Because this voltage will be exactly half of this voltage and this voltage here will also be exactly half of this voltage. So these two voltages are identical. Potentials at these two points are identical and therefore no difference between the voltages of these two points. So it is 0. If any one of the resistors is disturbed from this balance condition then a voltage will appear. Now what is shown on the slide is the ratio of the voltage that appears the output voltage and the input voltage as a function of the values of the resistors. Now there are two possible formulae shown here. So a person has done a derivation and somewhere he might have committed a mistake in the derivation. So at the end he has ended up getting an error in his formula. Now which of these two formulae is correct? Suppose if you want to check. Now one method is of course to go through the derivation and check where is the mistake. On the other hand this approach of solving problems by limiting approach or stretching to the extreme. So what is it that you do? Consider the problem for an extreme situation. Stretch the situation to an extreme in the present case for example. What is it that is causing the output voltage to differ from 0? It is the delta R because when the bridge is balanced delta R is 0. Let us look at the paper slide again. It is this delta R that is responsible for causing a voltage. Let us stretch the value of the delta R to the extreme. You will find the formula will reduce to a simpler number. Each of these two formulae will reduce to a simpler number. If I put delta R tending to infinity here then I will get V o by V i as 1 by 2 in this formula. Here however I will get V o by V i as 1. When delta R tends to infinity. So now I have to just compare these two cases for delta R tending to infinity which is very easy because if delta R tends to infinity that means this resistor has been removed. That is the meaning. It is an open circuit. If I remove the resistor now what is the value of the voltage? Clearly if I remove this resistor then whatever voltage is there appears at this point. On the other hand this side the voltage is V i by 2 whereas here the voltage is V i. Clearly therefore the difference in the two voltages is V i by 2. So V o by V i will be half. So clearly the formula on the left hand side is the correct one and the formula on the right hand side is not correct. So like this particularly in derivation of analytical formulae. One can even check complicated formulae particularly when the formula is complex. It is not a good method to go through the entire derivation and check where is the mistake. Instead one should stretch quantities in the formula to the extreme. Extreme does not necessarily mean infinity. It can be 0 also. Zero and infinities are the extremes and then one can check how the formula changes and what is the value it reduces to because extreme situations can be analyzed very quickly. So this is one more skill in problem solving. Let us summarize the various skills that we have discussed for problem solving. The various strategies of problem solving are representation, logical thinking, then division into sub problems. So combination of logical thinking and division into sub problems and representation is analysis. Analysis involves all these three. So we have seen an example of why the computer is down. You have drawn a diagram which is the representation of the problem which can also be regarded as a block diagram. Incidentally, there was a query that we have missed out block diagram. That diagram that we have drawn is in some sense a block diagram. So combination of representation, logical thinking and division of the problem into sub problems. Combination of these to analyze is analysis. Then stretching to the extreme and verbalization. That is talking about a problem is one method of getting solutions for the problem. That is verbalization. Although not shown here, I want to also point out one more aspect of problem solving and that is ability to estimate quantities. The word estimate is used when you want approximate values of quantities. So to list it with an example, what is the thickness of this A4 size paper? Supposing I want to know in some situation, this is the information that I need to solve some problem. What is the thickness of this paper? So what is the way you would estimate? So evidently you will put a number of papers together so that the thickness becomes something that you can easily estimate. So maybe a millimeter thick or something like that. And then you will find how many papers you have put together and then divide a millimeter by number of papers and estimate. Is there any other method? So this feel for various quantities. This is very, very important in problem solving. So how many students can be accommodated in this room? It's a problem in estimation. So like this, in fact, there are very many situations where you need to have an initial estimate of the situation. And then again, what is the advantage of ability to estimate? You can make simplifying assumptions to solve the problem. So the ability to estimate is a very important ability which is again not used very often by people. Then we discussed the techniques of representation in detail. Some of the techniques that we considered are symbolic representation. Symbolic representation and diagrams and so on includes block diagrams also, reformulation. Then table, drawing tables. So you have list form of table and matrix form of table. Then drawing graphs using bend diagrams and drawing diagrams to scale.