From the terror of Boko Haram and Aiswab in the northeast to the Rambeidah Bandits in the northwest, Aipop in the southeast and other militants elsewhere, Nigeria has suffered a crippling security situation. And on May 29, we know that President-elect Bola Mettinibu will not only inherit the mantle of leadership from President Obama to Buhari, but also he will be demanding a task of ending Nigeria's myriad of security challenges. Now in over 10 years, Nigeria's security landscape has been shaped by war against insurgents such as Boko Haram and later on the Islamic State of West Africa province Aiswab in the north eastern parts of the country. New threat elements such as banditry and kidnappings have emerged in the northwestern region while age-old communal clashes as well as farmers and hairdress clashes have existed in many parts of the north central. And also under President Buhari, secessionist agitations by the indigenous people of Biafra, Aipop in the southeast and parts of the south-south have also snowballed into a major security threat that is threatening the country's unity. Joining us once again to discuss this is Chief Dallington Momo Umoru. He is a former consultant to the Presidential Committee on Small Arms and Light Weapons Prescott. Thank you so much Chief Dallington for joining us again today to continue this conversation. My pleasure. Great. Let's talk about the major issue now that Nigeria is having to deal with. Like I said yesterday, we've heard all kinds of tough talking, declaring areas no fly zones and saying we want to put an end to. In fact, the Buhari administration has continuously held on tightly to the statement that Boko Haram has been technically defeated. Just as I said in my intro, there's several other issues bedeviling the country's security. Let's look at strengthening our security forces. Several millions, if not trillions of Naira, is earmatched to fight security, not to talk of security votes that are never made open and so nobody can account for it. How well has the Buhari administration dealt with strengthening our security forces? My first guess was talking about increasing the number of soldiers because this is what the President-elect thinks that will be the solution to the problem. But is that the only way that we can deal with our security situation in the country? Apparently not. There's no way you want to say that increasing personnel will end insecurity in Nigeria. Security in Nigeria has a widespread as it is today. Of course, you are aware that the level of insecurity in Nigeria differs. What you find in the Northeast is not the same thing you find in the Northwest. It's not also the same thing you find in the North Central and then it's not what you find in the South, like the Southeast, the South-South and the Southwest. So, apparently, one solution cannot bring to an end the entire level of insecurity Nigeria is currently suffering from. But be that as it may, the basic problem, like I started with yesterday, is the issue of the proliferation of small arms and light weapons. The amount of illicit fire circulating in the country, particularly in the hands of the non-state actors, is the biggest problem we have. All of these crimes, whether terrorism, whether banditry, whether armed robbery, whether courtism, name it, whether kidnapping, none of these can thrive without the availability of illicit arms and ammunition in the hands of these criminals. So, apparently, Nigeria is under, the Nigerian military, the security agents have, are other personnel. Because the view is, let's look at it from this point of view, research has shown that for every day, particularly every 24 to 48 hours, one police officer has gone down or one police officer is maimed. The same thing goes to other security agents and then the armed forces. And when we are talking about the armed forces here now, we are referring to the army, the navy and the air force personnel. Some of them are permanently stationed in the Northeast or well at the theater of war. Many of them don't come back, you've seen stories at the time, the situation is better now. Before the matching order given by President Boari in about October last year, what we used to experience on a daily basis was bombings, killings of other security agents or the military or even the civilians who are not even armed. But again, going by the UN resolution, Nigeria is under police. Nigeria, the military, of course, has a primary responsibility to defend the territorial integrity of Nigeria. There are not civil cases. So those that are combating today are terrorists in whatever guise, in whatever phase. You have them drafted to the Northeast, you have them drafted to the Northwest, you have them drafted to the North Central, you have them drafted to the Southeast, you have them drafted to the South-South, you have them drafted to the Southwest. Election matters you bring in the military. Intracommunial issues you bring in the military, intercommunal issues you bring in the military, tribal conflicts you bring in the military, religious conflicts you bring in the military. And so how can you have it? But why do you think that that's the case? Because you see, we know the difference, especially the fact that we have a president, sitting president, who's a military leader, who was a former military president, who knows the difference between policing and the job of the military. But why do you think that the military has now been taking over the job of policing within the country as opposed to protecting our borders? Why do you think that that's holding sway? Two factors, I'll give you two factors, for instance. Number one, Nigeria is under police. Even the Nigerian police as it is today cannot give you an exact figure of the numerical population of the Nigerian police. It's all about guesswork. They'll tell you 35,000. They'll tell you about 36,000. They'll tell you about 37,000. It is not their fault. It is so because number one, many of them are killed on daily basis or mailed. Number two, the military. The military are even those who train the police. They have special training. The military is trained far ahead of the Nigerian police. But unmarriedly, issues within the country are not supposed to have the presence of the military. But apparently because the police, we don't have enough of them. Even the total figure of the Nigerian police we have, there's also an issue. The police has what is called the SBU, the Special Protection Units. The Special Protection Units are used to protect the VIPs and the VVIPs. They are used to protect financial institutions and major other institutions. What we have annually or politically when you have a new inspector general of police appointed is that in most times at that point in time, they want to withdraw all the SBUs and ask them to return. But we get after what happens, those SBUs are sent back to the VIPs and the VVIPs. So how can you, how many of the police, how many of the 5,000 policemen, over 200 and 15, 216 million Nigerians? You don't have enough of them. Also when there's crisis, you have to bring hands from the military to assist the police, to quell down whatever companies the country is going through at that particular point in time. So the issue of personnel like he stated is very key, but that's not the basic one for me. The basic issue Nigeria needs because policing today has gone beyond the use of manual manpower, physical manpower. Across the world in developed countries, policing is done more with IT gadgets. I give you an instance. Look at what happened in Sudan, the crisis in Sudan recently. Where countries had to take away their citizens, right? America had to fly some of its citizens from Sudan. The rest of them that could not be taken away from Sudan were driven by road in buses, including the American Allied citizens were driven in buses and taken to over 800 kilometers stretch to another destination from where they found safety. Now, look at what happened. America didn't send physical soldiers or policemen to escort those buses for a stretch of over 800 kilometers. What America did was to deploy combat UAVs. When I say combat UAVs, I'm talking about combat on unmanned area vehicles. Talking about drones now. You remember the issue in Iraq and Iran and America sometime that created a very huge crisis where America took out two heavily one from Iran, one from Iraq. They monitored them from the Iraqi airport and then took them out with the drones without the physical presence of any military officer any American officer or Iraqi officer. Okay. And you remember that COVID-19 was what actually stopped, what would have become the third world war at that point in time. Okay, let me bring you back to the conversation. Sorry, because we don't have too much time. Let's talk about developing a national security strategy because well, we're also trying to set the agenda for the incoming administration. What should be the makeup of that national security strategy? Because if we keep having these issues of insecurity, it definitely one way or the other will affect governance as a whole. What would you be proposing? Well, the one of it is to revisit the security architecture as it is at the moment. What and what this government that will be living in a number of days did not get right. You want to look at one and the issues of arms smuggled into Nigeria. It has created a lot of devastation. The porous borders on the northern part of this country, particularly the illegal routes, have not helped the movement of illicit arms into this country. Okay. Now, the issue, like Yara Dua did, one of the basic things because this government really could not do or get right was bringing some persons to the table to negotiate with them, to discuss with them, to listen to them. For instance, the issue of iPop that is taking place in the southeast at the moment. The courts have asked that none of the cannons should be granted. But well, the security agents have their reasons for not releasing him up until this moment because apparently they are still doing their investigations and all of that. And the matter is in court. But the court has asked that the man should be granted bail, which is one of the demands by the Orhanese, the Orhanese indigo, which is also the demand of the governor of Anambra State who has also tried to wade into the crisis in the southeast. And then you also look at the problem of inter-gadrin. We really do not have enough gadgets for surveillance. We do not have enough drones, surveillance drones, surveillance UAVs, surveillance equipment that can also help in most other parts, like you have in the northeast at the moment. Equipment, do we have enough equipment to carry out all of this, what seemingly looks like war against terrorism?  And then we're also talking about corruption. Corruption in the system, corruption in the purchase of equipment. And then we also need serious bilateral relationship with some of these countries that manufacture this equipment. We must, most of them have reasons to sell equipment to this country. Some of us do not know. There's also the black market, where in most times some of the countries have to go to purchase equipment. We need to go to these countries and convince them that this equipment and because the fear, the fear they have often are laid is the fact that this equipment, if they are sold to us, will go into the wrong hands and then they will be misused. Even some Nigerians from some regions, forgive me, I will not mention those regions, have written to some of these countries in Europe and America to say that they should not sell this equipment to Nigeria because they will be used against them, because they are oppositional in whichever form. In consideration of all of this, we need to build that trust with these countries who manufacture this equipment. You remember the story of the Tucanos? Setting government came. What is government was in the one who started the issue of this Tucano, getting Tucanos from America from those countries. But it took them a very long time before they could agree to manufacture those specific Tucanos for Nigeria. And even when they sold those Tucanos to us, you remember what happened? They placed conditionalities on those Tucanos. They said those Tucanos must only be deployed against terrorists. If you remember all of that, that was why the hand of the military in Nigeria and the Air Force, particularly, were tied because those Tucanos could not be deployed against the bandits. Oconti, Nigerian government took that matter to court where the courts not declared banditry as terrorism. We thought that, ordinarily, you couldn't have been able to deploy. Now, these are some of the ways this international committee tied the hands of this country from really going all out to fight against terrorism and the bandits. But thank God, all of that has been solved. Many of you do not understand the reason why the activities of ESN has gotten to the extent it got to right now. You remember that ESN is the militant group of the IPOC. IPOC created that body because they said they needed that body to tackle, well, whether truly or firstly, to tackle bandits in the area that were infiltrating the southeast. The southeastern governors also came together to put up a security arrangement and architecture and the quality of a Bubeagu that never saw the light of day. Where is that security happening today? Now, in Nigeria, what you find in different places are things that look like regional security bodies put together by certain regions in the country. If you come to the southwest, you have a regional security body. That is because these bodies are created either because the people of this government do not really have full trust on our security agents and the military that are doing very well or because that they want to also assist with this regional security benefits. All right. In closing, because we have just one or two minutes, many trusted President Bohari, knowing that he was a former military leader, to deal decisively with the issue of insecurity, being that he also campaigned saying that he was going to put an end to it. So what are the chances that Ebola met Tinnabu is cut out to deal with this insecurity if his predecessor was even unable to deal with it in closing? Well, if you ask me, I think Ebola met Tinnabu is in Nigeria and has lived in Nigeria and has seen the level insecurity go to in Nigeria. In his campaign, he talked about rejigging the security architecture. I will take you back to what he did in terms of managing security in Lagos when he was governor of Lagos State. You remember all of what he put together, the lives of Operation Street, the lives of the kind of architecture that he put in place in Lagos that brought down crime dramatically, drastically in Lagos State as government. Today, that architecture that he put in place is still running in Lagos State. You see that in Lagos State, you even have what is called a security trust fund. Now, there's something that I do not really support. Whether creating security votes that governors are collecting on monthly basis has not done Nigeria any good. What are those security votes that nobody is even aware of the sum that governors collect on monthly basis are used for? In Lagos State, you could say, well, it is channeled to us the security fund. We have seen physically what Lagos State government, over time government or internal government have done with the security agencies by assisting with equipment, by vehicles, they buy all of that to support. That's why you find security in Lagos better than most other parts. In other states, where you really have the problem of insecurity, you have it in Benway State. Now, look at what is going on in Plateau, in the northeast and all of that. What are the governors, how are the governors supporting with their security votes? Are they supporting the military and security agents? That is one thing I think the incoming administration of Bolagme Tunugu should revisit and propose a bill to the National Assembly in respect of that. Is a bill ever passed, the security vote, so that those counties can be converted to the purchase of equipment for the military and the recruitment of more personnel? All right, I want to say thank you. Darlington Momo, who is a chief, is a former consultant to the Presidential Committee on Small Arms and Light Weapons Press Come. Thank you so much for speaking with us on this issue. What will be happening to security in Nigeria remains to be seen. Thank you so much for being here. My pleasure. Any day. All right. All right. Thank you so much for being on the show tonight. We want to thank you all for watching.  Just head to PLOS TV Africa on YouTube, like, subscribe and follow all our programs. I am Mary Anacone. Have a beautiful evening.


Welcome to JSA TV, the newsroom for tech and telecom professionals. I'm Jamie Scato-Cutaya of JSA here at ITW 2019. Joining me today, Mr. Dan Flatt, he's the Director of Marketing for Mercure. Tell us a little bit about Mercure. Thanks, Jamie. So Mercure, we are a proudly Canadian company. Our special ingredient is mesh Wi-Fi router systems. So we have our DNA in the Waterloo tech ecosystem where BlackBerry was born. And actually our Chief Scientist was one of the original engineers at BlackBerry. So we know how to do antennas and transceivers, the building blocks of mesh Wi-Fi systems really, really well. And that's been one of our main advantages in the market. Oh, I love that. And I know you're very proud to be Canadian, but you also are global and operate in so many countries. Tell us more. Yeah, so we actually have our products in about 47 different countries right now. We've been able to accomplish that both by selling directly to consumers. We had a crowd-funded model launch directly to consumers last year with our current generation product. But we've also had the opportunity to work with a lot of partners around the world, internet service providers and other carriers. So they're able to provide our products directly to their clients and in countries like Australia, Benelux, Israel and others. Wow. And I'm so impressed by this vantage point that you can bring to the marketplace, this unique perspective. Can you talk to us, for example, on what you see in terms of IoT adoption on a global scale? Absolutely. So I think globally the conversation around IoT has really been focused on what it can do for enterprise, for commercial enterprises by making systems more intelligent, creating interconnectivity between these networks so they can be more efficient and save money. But I think that for everyday people, homeowners and end users, they're not really experiencing IoT to the same extent unless it's really coming to them in their homes. And that's why for us, the conversation around IoT, we're really trying to focus it on the smart home. Yeah, and that's a great point. So your end users, your homeowners, how do you define a smart home? Yes, a smart home, it's become a little bit of a buzzword, like a lot of industry terms. So if we can break it down, the home hasn't really evolved beyond the four walls, maybe sometimes a few other walls and the roof above. As much technology as we will pack inside that and cohabit with that technology, really making the smart home smart is allowing it to think in some ways and even make decisions. And that means implementing IoT into the home, into all the different systems, whether it's water systems, electricity, HVAC and data as well, so that we're able to let the home actually have a nervous system and we're providing the brain for that nervous system with our mesh Wi-Fi systems. I love that analogy and I believe I got to chat with him ahead of time, of course, and I believe what you had said to me yesterday was that you have the capacity for up to 60 smart devices in one home. Absolutely, yeah. So we can expand, the beauty of mesh Wi-Fi systems is that it's not a one size fit all solution, we have two different types of units, queen routers, we use the terminology queens and bees and we call our whole system a hive. So queens can cover about 3,000 square feet, the bees can cover about 500 square feet, so you can really tailor a system to whatever home you need and in a lot of countries with different home sizes, different construction of the way data signals will permeate through walls that are of reinforced concrete, for example, we can really tailor the solutions to every type of home so that you can connect whether it's just, you know, your laptop in a tablet or you've got a smart thermostat, you've got a Roomba and all of these new fangled wireless devices that are connecting to Wi-Fi. Yeah, I can't wait till my fridge tells me when my milk's expired and orders me a new one. It's not doing that already? I am, I'm behind the times, behind the times. And here at ITW, bringing it back home to our fabulous event that we're here at in Atlanta, there are so many global ISPs, carriers walking the floor. How do you work with them? What are your benefits that you bring to the table? Yeah, so we've seen the conversation at ITW really shift over the last 10 years or so from voice over to data. And I think when we walk around and talk to a lot of the companies here, a lot of the focus on data is laying the infrastructure, getting that last mile delivery service. We're really focused on the last few feet. And to use another analogy, if you say that companies are doing a really good job of laying the pipes and enabling that fire hose that is gonna reach every home, we're the sprinkler system that's gonna distribute that the data signal, if data is water, to every corner of where it needs to be. And so we're really interested in connecting with carriers, ISPs, anybody who is enabling homes or small businesses with data connectivity to provide them with the hardware that allows their end users to fully enjoy connectivity in every corner of their home or building. Thank you so much for joining us. It was a pleasure to learn more about your company. This is amazing. And thank you viewers for tuning in to JSA TV here at ITW. Happy networking.


So, welcome back to the continuation of the previous session. As I mentioned, I will very briefly discuss what is titled here as Hemchandra's problem. Actually, it should be renamed as Virahanka problem. I will mention the reference later. The objective of this particular example will become a clear as we discuss this example. Please note that what I have been trying to emphasize is that teaching our students the basic skills in control structures and data structures is important and critical. But once they learn those basic skills, what exactly to do with those skills, what programs to write depends upon the problems that we wish them to solve. Each problem should be so chosen that it illustrates not only the proper translation of algorithm into a program, but it also relates to some interesting real life problem which our students can relate to. So, here is an example where I start with a problem which is both real life as well as slightly artificial. The real life connection is established through an example of talking about building a wall. A wall is built using bricks. We will convert it into some kind of a permutation combination problem and then try to see how this problem is solved. This particular problem of course has a greater relevance as I mentioned at the very beginning. This is the first time we will talk about a recurrence relation to our students. So, here we go. Suppose I have to build a wall of length 4 feet. Suppose I have bricks of only 2 sizes. One brick which is 2 feet long and another brick type which is 1 foot long. I have a large supply of 2 feet bricks and I have a large supply of 1 foot long bricks. The question that is being asked is that if I have to build a wall of length 4 feet in how many ways I can lay the bricks so that I can fill the length of 4 feet. This is very simple. The length is 4 feet. There are 2 feet bricks and there are 1 foot bricks. How many different ways in which I can fill up this length can be easily imagined by our students? We write those possibilities. One, there can be 4 bricks of 1 foot long. So, I cover the length 4 feet. Second, I have a 2 feet brick and 2 bricks of 1 foot each. Third, I put 1 foot brick first, then a 2 feet brick and then a 1 foot brick. Fourth, I put 2 1 foot bricks first and then put a 2 2 feet brick. And lastly, I can choose 2 bricks of 2 feet each. So, these are the different ways in which I can use 2 types of bricks 1 foot long and 2 feet long to fill up the length of 4 feet. Having done this, suppose we ask the question in how many ways can this be done if the length of the wall is say 3 feet or if the length of the wall is 5 feet or the length of the wall is 8 feet. Now, individually for any length that we have, people can work out all the possibilities. However, our aim is to illustrate that there is a certain pattern in the way the bricks can be used to fulfill any given length. This is explained in the next slide here. These are the possibilities if the length of the wall is 3 feet, 4 feet or 5 feet. So, here are the possibilities written if the wall is only 3 feet long. One, I can use 3 bricks of 1 foot each. Second, I can use 1 brick of 2 feet first followed by 1 brick of 1 foot long. Third, I can start with 1 foot long brick and then add another 2 feet long brick. So, there are exactly 3 possibilities if I want to build a wall of 3 feet length for laying down the bricks. 4 feet wall we have already seen. I have repeated all the possible ways here. If I were to have a 5 feet wall, then I can write the various possibilities here. Notice the possibilities are written in a peculiar fashion. What is this fashion? The fashion is that I notice that for example, if I have to build a 4 feet wall, then the number of possibilities are worked out by first taking all the possibilities in building the 3 feet wall, repeating them here and then adding just a 1 foot break for each of the possibilities. So, there were 3 possibilities in which the 3 feet wall could be built. I take each one of those possibilities, add 1 foot break and then I will get the possibilities, some possibilities for building a 4 feet wall. This is natural because I have a 3 feet wall. If I add one more foot to it, it becomes a 4 feet wall. So, it is natural that I add 1, 1, 1 here and I will get a 4 feet wall. But additionally in a 4 feet wall, I have 2 more possibilities. I can have 1, 1, 2 and I can have 2, 2. That means I added 2 more possibilities in which the last brick is 2 feet long. In all the first 3 possibilities, the last brick was 1 foot long. This is the crux of the observation by Hemchandra and even before that by Virahanka. I will explain that later. But let us look at this in a more systematic way. So, let us go to the next slide. Here it is shown that the possibilities for building 4 feet wall are partially determined by the possibilities of building a 3 feet wall and by adding one to it. In addition, I have 2 more possibilities. Now, if you notice carefully, these 2 possibilities emanate out of considering the possibilities of for building a 2 feet wall only.  So, there were 2 possibilities. For each of these 2 possibilities, if I add a 2 feet brick now, I get a 4 feet wall. So, 4 feet wall essentially can be built in how many ways? 1. As many ways, I can build a 3 feet wall by adding a 1 foot brick and plus as many ways, I can build a 2 feet wall and add a 2 feet brick. Now, I extend this further to 5 feet wall. So, if I go to the next slide, I have shown it here. What is this 5 feet wall? The 5 feet wall contains combinations which were all there in the 4 feet wall. For each of the possibilities of building the 4 feet wall, I take all these possibilities and add one more brick. I get a 5 feet wall. Additionally, if I take the possibilities for building the 3 feet wall and put them here in addition to the earlier possibilities and add a 2 feet brick, then I will get a 5 feet wall. This leads us to the conclusion that the possibilities for 5 feet wall is the possibilities for 4 feet wall plus 1 brick added and plus possibilities of 3 feet wall plus 2 brick added. What it means is, if I just look at the total number of possibilities, then the total number of possibilities are simply the number of possibilities for building a 4 feet wall plus the number of possibilities for building a 3 feet wall. If I add these two possibilities, I get the possibilities for building a 5 feet wall. The actual problem which Virahanka and later Himchandra Tackle was slightly different. The Virahanka's actual problem was that of building a poetic meter. Now, I do not know how many of you are familiar with poetic meter, but it is suffice to tell our students that a poetic meter is made of short syllables and long syllables. So, a long syllable l has 2 beats and a short syllable s has 1 beat. Please note now, people can relate this 1 beat and 2 beats to the 1 foot long brick and 2 foot long brick. Just as an example of a poetic meter, you can take any one of the standard poems or songs that you come across. Here is an example that I have chosen from something which is well known in India. This is part of the Saraswati Vandana, Ya Kundendutushara Haara Dhavala, Ya Shubhra Vastravrata. I have written this line and I have shown the syllables which are used in this particular line. So, Ya Kundendutushara Haara, these are all long syllables, Ya Kundendutushara Haara, like that. So, you have long, long, long, short, short, long, short, long, short, short, etc. And this particular sequence is an example of a poetic meter. Now, the question that Virahanka raised is, if I am designing a poetic meter with 8 beats, how many ways are there of filling 8 beats? You will appreciate that my student who has first seen the example of Brex and which he or she can relate easily to, will find it very simple to extend that example to the concept of determining the possibilities in which such a poetic meter can be constructed of as many beats as we prescribe. Here, we are saying that if I am designing a poetic meter with 8 beats, how many different ways of filling these 8 beats are? The interesting solution by Virahanka was to simply say that if I follow the method of Pingala, Pingala was a person who lived in 200 B.C. There is a mathematician poet of that era. Then, according to that method, it is enough to observe that the last beat is long or short. Very interesting. He is saying the number of possibilities can be determined by merely observing whether the last beat is long or short. Let me go back a few more slides to emphasize this point from our example of the Brex. If I want to determine how many ways a 5 feet wall can be constructed, I have to write all the ways. If I observe whether the last brick is 1 foot long, that is short brick or the last brick is 2 feet long, which is the long brick. If I combine the number of ways in which the last brick is 1 foot long or effectively for poetic meter, the last beat is short. All the possibilities where the last beat is long or the last brick is a longer brick, then I automatically know the total number of possibilities for the 5 feet wall or a 5 beat meter. Let us go back to the next slide here. Here is another point that Professor Abhiram Ranade had stressed during his lecture. Although I find that he has not emphasized it in his textbook, but please notice that Hemchandra is giving credit to someone who lived more than 1000 years before him. In Hemchandra's writing and also in Virangaka's writings that have been since, both mentioned the method of Pingal, that means they are giving credit where it is due. Moral for us academicians and this is interesting and useful to emphasize to our students copy if necessary and if permitted. Pingala, Virangaka and Hemchandra were obviously the firm believers of open source knowledge. So, they have left all their knowledge without any incumbency of intellectual property or whatever anybody can use it, but it is correct. So, therefore it is permitted. However, it is important that when we copy, we always give credit. I think this is not related to programming, but this is an important lesson that we should all like to empathically tell our students to internalize. Anyway, let us go ahead with this discussion. So, the original solution said that an 8 bit pattern can be made of take any 7 bit pattern and add a short bit or take any 6 bit pattern and add a long bit. Very obviously, then the total number of possible 8 bit patterns is simply the number of 7 bit patterns plus the number of 6 bit patterns. In short, the number of 8 bit patterns is equal to the number of 7 bit patterns plus the number of 6 bit pattern. I hope now all of you are realizing what I am leading to. Algebraically, if h n is the number of patterns with n beats, then h 8 is equal to h 7 plus h 6 and in general h n is equal to h n minus 1 plus h n minus 2. Anybody who can recognize this relation does this help us to compute h 8? Yes, provided we know the value of h 7 and h 6. Well, does this formula help us to calculate h 7? Yes, provided we have the value of h 5 and h 6. This is a beautiful way of telling our students of what a recurrence relation is. A recurrence relation is built step by step starting with some initial value. The recurrence formula is simply h n is equal to h n minus 1 plus h n minus 2. Let us use one more slide to explain to our students how do we calculate successive values of h n. So, we say we start with h 1 which is the number of patterns with 1 bit which is just 1 say short bit. h 2 which is number of patterns with 2 bits which are 2 short, short and long. h 3 which is the number of patterns with 3 bits which as we saw from the recurrence relation will be h 2 plus h 1 which is 2 plus 1 and we can verify that indeed there are 3 possible patterns short, short, short, short, long, long, short. Subsequently when we tell our students this formula we need not elaborate and write down all the possible combinations, but people will understand it. What is more important from the point of view of learning to program this problem is to illustrate that h 4 is equal to h 3 plus h 2 which is 5, h 5 is h 4 plus h 3 which is 5 plus 3 which is 8. We have already given a sample illustrative example figuratively for these 5 and 8 ways by the example of all the bricks. By now students will very clearly understand that if I continue evaluating this recurrence relation for h 6, h 7 finally I will get the value of h n. We now know that my dear students to calculate the value of h n we need to use two previous values. By using any two previous values we can calculate the next value. Now this is a recurrence relation. Some of us might want to elaborate something more about recurrence relation at this stage. Some others like me would say that we will worry about the notion of recurrence at a later stage, but at this juncture we will just note that h n is equal to h n minus 1 plus h n minus 2 is a recurrence relation. However, we would like to use an iterative method to solve this problem. In this iterative method we can start with two values and then successively calculate the subsequent value. Now comes to the program design crux. We know the algorithm already. We need to use two previous values to calculate the next. We want to use an iterative solution within the iteration. Every iteration will do the following. It will use the current and the previous values to calculate the next value. Now that is not sufficient because whatever variable we store the current value in and whatever variable we store the previous value in will retain the same value if we blindly come back and iterate around this again. What we need to do is, before going to the next iteration we have to reset the current and previous value. So, whatever next value we have calculated must become the current value and whatever was the current value earlier in the loop must become the previous value. So, that is why I have stated reset these appropriately before the next iteration. And there is an additional comment made. Start with the first two values h 2 and h 1. You can give an example. I have written here in the slide that you can say h current h brave h next. By the way these are clearly the intuitive ways of naming the variables. Of course, people can use any names that they choose, but here I have chosen to use the words h current and h brave for the previous term and the current term. Using these I will be calculating the next term. I have written here starting values. You may use the white board or a transparency or whatever in your lecture, but do consider the following. Write down a set of at least some initial steps here. So, first iteration calculation. It is best for you to use such a sort of drawing on the board and write down physically against this what will be the value of h next. It will be simply 2 plus 1 that is 3. Now, against this reset the two values what you have to do is you have to say the value 3 becomes the h current value, but before that the value h current goes into the h brave. So, when I go to the second iteration I would have h current as 3 and h brave as 2. This resetting has to be explained to the students. As I said you can use either a slide and write on the slide or you can use a white board where you actually show demonstrate how things change in every iteration. I would like you to appreciate that while you and I as teachers who would have known the iterative solution for recurrence relations many times over, we will assume that resetting the two values is something which will happen automatically in our minds. Unfortunately for the first year students it is not very easy for them to understand that there are two variables whose values need to be reset before you go to the next iteration. That h proof must be reassigned the value of h current and h current must be reassigned the value of h next before you go to the next iteration is important. It is therefore useful to always show this draw the diagram and write down the values as they change. It is only after this that we can go over to write the program to compute h n. Interestingly the program is extremely simple. It says collect the value of n which m channel number you want to compute whether it is h 5, h 8, h 10 or whatever. Then I start with the initial values int h proof and h current. h proof is assigned the initial value 1, h current is assigned the initial value 2. I also define the variable h next which at any point will be the next him channel number calculated. I set up an iteration now. Please notice that I cannot calculate the first and second number by using that relation. These are predefined values which are assigned by knowing that the first value is 1, second value is 2. Very obviously if somebody gives an input which is 1 or 2 the program will not work correctly. The loop says for int i equal to 3 i less than equal to n i plus plus. So the first iteration starts with i equal to 3 then it will be i equal to 4 and so on. It simply writes down the assignment statement h next is equal to h proof plus h current. That is all you have calculated the next n channel number. But if you want to go to the fourth and the fifth and the sixth you will have to reset those values as I said. Here is a comment which says prepare for next iteration by shifting values of objects which are these objects h proof and h next. So I say h proof equal to h current and then I say h current is equal to h next. Having done that this iteration will continue till n. When I come out it is very obvious that the last value calculated of h next will be the one which will be the nth Hemchandran number I output that. So notice that even though the program is only a five six seven line program for many students this logic of resetting the values before the next iteration will be slightly cumbersome. They would have done similar things earlier in the context of some other example. But in the initial days when we engage the students in their understanding iterative solution it is useful to emphasize that such resetting will be almost always a basic feature of most iterative solutions that we set up. In the earlier problem the mere change in the value of I took care we did not have to reset anything. But in a majority of the problems we will have to preserve the value that was computed in the previous iteration as the starting value for the next iteration for some purpose or the other. In this particular case there are two such values. Professor Ranade calls this mathematics from poetry and I agree with him. This interesting series has been since well-known and the reason why it has become well-known is that a European scientist extended this series to many interesting usage. For example number of petals in many flowers ratio of consecutive terms of this series tends to a limit which is known as golden ratio. Now all our students and all of us know these more commonly as Fibonacci numbers. Hemchandra and Virahanka who actually given the Pingal Sutra solution lived before Fibonacci and Pingal lived several centuries earlier. I like this example for another reason. It is only recently that the correct attribution to the original discovery of Fibonacci numbers has been given back to the Indian origins. If you look at the Wikipedia article it clearly mentions and in fact there are several recorded historical documents which indicate that Fibonacci himself admitted that he learnt about this interesting series from some Arab scholars when he was visiting Arabia. And that several Arab scholars mentioned to him that the basic theme for these sequences they learnt from some Indian scholars. That is how the history has been traced back and has been established. Nothing very big about it and nothing much related to computer programming really. But it is a point that we should make to our students that while much of very useful knowledge has come out of innovations, discoveries and inventions from the developed countries. There are several important Indian discoveries which ought to be remembered by us rather proudly. I will stop at this juncture. There are some hands raised. Let me go over to Dronacharya College of Engineering. Yeah, I have given this solution of bi-state description of patterns which you have just discussed through Fibonacci series. Fibonacci series is generally the bi-state pattern descriptor which depends on two of its previous states. And whatsoever the two previous states are if we append those two states then we get the current one. That is the generator of Fibonacci series generator. Very good. Thank you so much. Let us go over to some other institution. Is Vivekanan College of Engineering? Sir, I am Malathi from Vivekanan College of Engineering for men and women. I am interested in the relationship between the Dutunaki series and the Hamachandra series. Oh, okay. The relationship is very simple. They are identical, exactly the same series. In fact, this is what I tried to explain. Hamachandra sequence or Virahanka sequence as it is now more correctly called is nothing but the Fibonacci sequence. And in fact, the Fibonacci sequence is not the right name as has been established now. It ought to be known as Virahanka sequence. So, there is no question of any relationship. They are exactly same. They are identical. It is just that the name Fibonacci is more popular because that is what got printed and circulated everywhere. But it has been established that the initial discovery of this series happened in India. That is all. The name is different, nothing else. Okay. There is a hand raised by a C.S. graduate school of teaching. I am Aparna Vandoru and he have total seven participants over here. And the question is, can we use this, the first example which you have showed us finding out the logarithm. Can we use it for double, triple, integral and extend simultaneously for first year students to explain? Interesting question. So, my answer is as followed. Yes, I can actually extend it, but the issue here is not so much to solve complex integral calculus problem. The issue here is to explain to first year students how to do approximation of an integral value and how to implement that using a computational procedure for which we can write programs. My suggestion madam would be that we should spare our first year students in very complex mathematical theory that they will learn in due course of time. But in the subject of computer programming, while it is important as you point out to relate to the issues of integration and so on, but the emphasis must remain on translating such approximation methods into computations and to write program. So, I would suggest avoid double integrals or triple integrals in the first year level. In fact, students will get completely confused with the integration process rather than concentrating on programming. That is my view of course, but the question is well taken. Thank you. Let us go over to another question that has been raised here. SVP Engineering College. Sir, we would like to know about a particular aspect as far as the university syllabus is concerned for the first years in our college. We have numerical methods as well as computer orientation being written in C programming. How better it is to make student ease with numerical methods along with programming? First of all if that is the university syllabus then it is necessary to fill that up. Our own opinion in IIT is that in the first year you should teach the basic skills in programming and in the second semester courses like numerical analysis etcetera could follow. Traditionally please remember that numerical analysis is a subject which has been taught even before there were computers and quite often it was taught at the elementary level. That is how it has remained perhaps as a first year first semester course. If that numerical methods course is there and the computer programming is also taught perhaps some coordination by two teachers will ensure that whatever numerical methods are covered in the numerical methods subject are also used by programming course teachers to implement the corresponding computer programs. My own opinion is that a course in numerical methods using computers should follow a programming course. As regards the other part of your question that C program is mandated by the syllabus I would urge you to work with your university not to a particularly state that only a particular language like C should be used. Professor Ranade has already mentioned the problems in using plain C programming because you have to start with scanf and printf which people will have to just mug by heart because they really cannot understand why pointers are being used here and what is the method of formatting string when they do not even know the string. Please also note that all the examples that we use in IIT Bombay are actually C programming examples. We do not use any C plus plus specific construct in the entire course except towards the end when we introduce the notions of object oriented programming. The only syntax that we have used which is non C syntax is C in and C out. I will take your point in one of the sessions here we had used some macros to define some very simplistic input statement such as input integer or output integer which would map into a corresponding scanf and printf statement. The major difficulty in teaching C programming in our opinion is the clumsy syntax. C was never designed as an introductory programming language but C has come to be used as an introductory programming language. It is my personal wish and the wish of almost all faculty members in IIT Bombay who have taught programming that universities will now define a syllabus for first year programming course which they will generically say C plus plus. However, the syllabus will make it clear that in the initial period of the course only procedural programming should be taught and the object oriented programming should either be a matter of a second semester course which is there in many universities or could be very briefly introduced in the first course. Sorry for this elaborate answer but I understand that this is an important point but I would still suggest that much of the material that we are generating in this workshop effort is usable by the teachers at least partly by appropriately modifying the computer program if you have to insist on C programming in your college. Thank you so much. Now we will stop at this juncture.