She only wanted his money, so he got payback on her wedding day. Dino felt his heart beating rapidly inside his chest. Why did she do this? How could she break his heart in such an awful way? He wanted to marry her and have kids one day, but it seemed that his dreams were crushed along with his heart. The sadness he felt soon turned to anger. He wanted to get back at her. She wouldn't get any warning or compassion from him when he planned his scheme. She was going to get what was coming to her. Dino worked a tough job on an oil rig. Even though rig workers don't need a fancy education, the men that choose to work here are highly skilled. There's one thing more important than education, and that's experience. Dino had what the company was looking for. He noticed the small details while on the job and knew what to do. From the outside, people thought Dino had it all, a good job, a great house, and some really good friends. Dino had been lucky with his employment and his house, but he had always wanted something more. He had always had bad luck when it came to love. That would all change though. Some people would love a chance to be with Dino. He was clever, made good money, and was responsible. He was also very humble, and didn't brag about what he had. After saving up enough, he paid for a beautiful condo in full. The only thing Dino felt he didn't have was a girl that he could share his joy with. What Dino didn't realize was that the next woman to come into his life would try to destroy it. Dino's co-workers would have a new girlfriend every shore leave. Dino respected them, but that life wasn't for him. He wanted to find someone truly special. Dino may be a very patient person, but that doesn't mean he's indecisive. One day, his friends couldn't believe it when they saw the woman holding Dino's hand. When Dino met this woman, he was enthralled. Andrea was a beautiful woman who looked like a movie star. She could have easily been on the cover of Vogue. Dino's co-workers were also speechless when they saw his new girlfriend. But not for the same reason as others. A lot of Dino's friends were happy for him when he found Andrea. Others had their suspicions. One friend in particular named Josh thought she was the most selfish, attention-seeking woman he'd ever seen. Dino couldn't have known this at the time, he couldn't see the future. But warning signs soon started appearing. Sometimes, Andrea's self-centered side would show. She explained that it was due to a bad childhood including parental divorce. Dino was trusting and understanding, so he dismissed her behavior. Andrea was very excited to move into Dino's apartment. In the beginning, their relationship couldn't have been better. But while he was out on the oil rig he had a funny feeling. Dino started getting bank statements he couldn't explain. Andrea had his card so she could get the necessities while he was at work. But the grocery bill exceeded $300. And another $200 just a few days later. She couldn't have been eating through $500 in just a few days. She must have been getting cashbacks from the store and pocketing them. Dino had a very bad feeling. Dino had suspicions now. Her behavior made him feel like she might have been unfaithful. With his money in his condo, she could easily cheat on him while he was away.  He came home early to find something shocking. That's when things got worse. Andrea didn't expect him home for a couple more days, so when Dino entered his house, his home that he worked so hard for, it caught her completely by surprise. For Dino, it confirmed the worst suspicions he had of the woman he once thought was the one. Dino could feel his heart crumble into a million pieces as he walked into his bedroom. The woman he wanted to marry and grow old with was indeed cheating on him. But Dino kept his cool. He didn't lose his temper, he just told the guy to get out. He then told Andrea to get out too. He handed her the key to a hotel room, $50 for cab fare, and told her to leave. But Andrea wasn't leaving.  This ended up with Dino telling his security guard to call the cops. Andrea wasn't on the lease, so once the police arrived and verified who owned the home, they removed her. That should have been the end of it, but Andrea was vindictive. Dino had taken away a lot of the comforts Andrea had become used to. His cash, his card, and his home. Andrea had to face the music. Dino wasn't going to provide for her any longer. He took all of Andrea's things, and had them placed in a storage unit. But Andrea wasn't going to just sit around and be snubbed in that manner. She wasn't going to let him leave her in the dark. She was going to hit Dino where it hurts. Dino's pride and joy, aside from his condo, was his white Chevy Monte Carlo. Several weeks after the breakup, Dino returned home from another stint at the oil rig. Instead of finding his car in tip-top condition, he found something much worse. For flat tires, a smashed windshield, and a can of red paint had been poured over it. All of Dino's friends wondered who would do such a thing. But Dino knew the answer. Andrea had just done something that Dino would never be able to forgive. While most men would hit back where it hurts immediately, Dino decided to be patient. He knew that the perfect time to strike wasn't now. Dino returned to his routine as if nothing had ever happened. But he hadn't forgiven or forgotten what Andrea had done to him. Soon, he would find an opening, somewhere to strike back at his ex-girlfriend, and he would do it hard. Dino would never forget what she did, so when he learned that she was getting married, he started plotting. He learned of the engagement through a friend of his, Nancy. Her sister, Kathy, happens to be very good friends with Andrea, but Nancy knew the real Andrea and passed the information on to Dino. She would become a co-conspirator along with him, and together they made sure that what was supposed to be the best day in Andrea's life was the worst. Thankfully, Nancy knew exactly what type of person Andrea was. When she received the invitation to the wedding, the first thing she thought about was how Andrea had betrayed her good friend, Dino. She took advantage of the fact that he had property and made good money, mooching off him as she messed around with other men. Dino started off with an anonymous letter to Andrea. About a month before the wedding, Dino sent an anonymous letter in the mail. In the envelope was a picture of Monte Carlo, a wedding dress, and a ketchup packet. There was a single sheet of paper too, with three words, red on white. If Andrea had thought Dino would just turn the other cheek, she was mistaken. He hadn't forgotten what she had done to his heart and his car, and neither had she. He was going to get his revenge, and with this mysterious message, she knew it was coming. The note was enough to make Andrea worry about her pure white wedding dress. Nancy had given Dino a little piece of information that she had heard from her sister, the name of the store where Andrea was getting her dress. Dino went to the store and asked the guy if he wants to make an easy hundred bucks. All he had to do was call Andrea and tell her that they need her to come to the shop and choose the red lace trim for her gown.  The guy said he got a call from her fiancé and ordered the red lace for the dress. Andrea knew that it wasn't the fiancé, but she had a pretty good idea of who was behind the sabotage. She was freaking out with good reason, his staining her white dress red wasn't a matter of if to her, but rather a matter of when. She was spooked. With the wedding soon approaching, bright to be Andrea was freaking out. Dino waited until there were just a couple of weeks left until the big day. Nancy slipped up and told her sister that a friend of Dino's worked at the company hired to cater the food for Andrea's wedding. Andrea flipped out as soon as she heard this. She immediately deleted the pasta dishes and had tomatoes removed from the salads in fear that Dino's non-existent buddy might dump a plate on her. This wasn't where it would end though. Dino had about 30 different red lipsticks sent to Andrea's house from a variety of Amazon sellers, all paid from prepaid credit cards. She wasn't able to trace them back to him, but she knew it was him. Dino then had a red Sharpie placed under her windshield wipers for her to find one fine morning. By the time of the wedding day, she was a complete wreck, expecting to be sprayed by red paint at every turn. It was supposed to be the happiest day of her life. In reality, it was in Dino's interest for the wedding to be really enjoyable for all of Andrea's guests, just not her. At the wedding reception, Andrea was a basket case. She demanded that all the red flowers be removed from the centerpieces on all the tables. She refused to get within 10 feet of anyone holding a red drink. She spent more time and attention looking for the ambush, she was certain was coming, and Dino didn't even do a thing. Nancy witnessed everything. He'd turned her into bright as illa, and Nancy kept him updated with a running commentary as the night progressed.  In the end, the only thing stained was her reputation.


blew through the Jewish communities in Germany. The first crusade hit the Jewish communities hard. It brought suffering, it brought torment. And this trauma created Ashkenaz as the new practices of remembering. The old treasure of prayers was joined by a troubling prayer, which is part of the Jewish cycle of the year, the East Corp prayer. In its desperation, in its pain, the Jewish people addressed God in order to transfer this commemoration to God. As if they were to say, God, remember the things that we cannot forget. Ladies and gentlemen, as president of the state of Israel, the state of the Jewish people, I would like to open my address with the East Corp prayer, dedicated to the memory of our brothers and sisters who were killed, slaughtered and annihilated by the Nazis. Please rise for the East Corp prayer. Remember God, the names of our brothers and sisters Israel, slaughtered and annihilated by the Nazis. The names of six hundred thousands of people of Israel.  And the old meetings that were fought against the Holy God. Remember God, the name of their church. The church of the Holy God, Israel and the people of Israel from all over the world. And in the end, they were killed and killed. The victims and the victims of their lives and their death did not die. And we were in peace with them and said, Amen. President of the Federal Republic, my good friend Walter Steinmeier, and his wife, Elke Budenbender, Federal Chancellor Olaf Scholz, President of the Bundestag, Bebel Bas. Thank you very much for your moving words. President of the Bundesrat, Budo Ramelow. President of the Federal Constitutional Court, Stefan Habat. Dear representatives from all walks of life here in Germany. Ladies and gentlemen, I stand before you, but I do not stand alone. I stand as an envoy, as the President of the State of Israel. A sovereign, democratic, national state of the Jewish people. And the realization, the embodiment of the prayers of numerous generations. I stand here as a proud son of the Jewish people. I stand here as the representative of the ninth generation of Rabbi Hilman, who was a Rabbi of a community here in Germany. He was a successor to Rabbi Katzner from Hamburg and the surrounding communities. I'm here as the President of the State, and I'm here with many dreams and crises, with pain and healing, with destruction and independence. That are inextricably woven. And I'm here to share a message, to share a commandment. Apart from the Ten Commandments, and I love you next as you would love yourself, is maybe the most moral and the most obligating commandment for every Jew. And that is, remember, the Jewish nation is a nation of remembrance. Remembering is an indispensable part of our identity. When I step on German ground, I have memories. I'll take the picture album of my people from my backpack, and there's countless images of this country. Pictures of summits, pictures of the deepest abyss. And this is what I want to talk to you about here today. I want to talk about the past and the future. And I quote, there's a stone set on my desk. It says, Amen. It's a triangular piece, a fracture of a wall of a Jewish cemetery. Destroyed a thousand years ago in the city where I was born. A single word, Amen, etched into that stone. It's hard, it's definite, it's a stone, witnessing everything that ever happened and that will never happen again. May it be like this, Amen. These are the words that Israeli poet Yulam H. He wrote. He was from the city of Jerusalem. He was born in the Bavarian city of Wurzburg as Ludwig Pfeiffer. And it seems that with these words he wants to navigate between pain and pride in this country and he wants to bridge this chasm between his place of birth and his homeland. Throughout hundreds, thousands of years, Germany was a homeland for our people. It was comfortable. It was creative. It was a homeland where the Jewish people could prosper in culture, in spirit, in philosophy, in politics, in science, in many other areas. The leading rabbis, the most influential rabbis in the history of our people, grew up here in Germany. This is where Rabbi Hashiaki grew up. The most significant rabbis of the Jewish people were selected. Rabbi von Rotenberg and many others called Harry Shereem, the first ones. Rabbi Samson Hirsch and other members of the last generation. Germany was the home of many outstanding people of Jewish culture, spirit, the culture. Joseph Mendelssohn, Joseph Geiger and many others. Albert Einstein, Paul Ehrlich, Emi Nötter, Berthard Auerbach, Kurt Weil, Elze Laska-Schüler. And here we have to mention the pioneers of social work, Sidi Wonsky as Rosengu, and the influential Zionists who grew up in Germany and were able to flourish here, maybe in Swedish Karussia, Moses Hess, Levaburg and many, many others. In all walks of life. All of these people were born here, they were educated, they were trained here, their personality was created here, they made a contribution for the Jewish people to prosper, for the German nation to prosper. So this country is their country in the deepest meaning. It is embedded in our DNA, the DNA of my people. But, ladies and gentlemen, this is no secret, this land, Germany, was the place where the greatest atrocities were committed, that the Jewish people had to suffer. It started with pogroms, uprisings, massacres, destruction of entire Jewish communities that recurred over thousands of years, the way up to the deepest abyss of humanity, the Shoah. The history of mankind knows no comparable example to the atrocities committed by the Nazis and their henchmen. Never was a country like Germany responsible for the loss of all humanity, for the loss of all compassion. They were responsible for the intention to cruelly exterminate an entire people. My father, my recipes, the sixth president of the State of Israel, was among the commanders who liberated the concentration camps from the claws of the Nazi animals. And I will never forget the words he used to describe what he saw. The skeletons in striped prison suits this hell on Earth. People jammed together in barracks. Like you said, Madam President, he traveled to Germany in the year 1987 as the first president of Israel and visited his visit in the death camp of Bergen Belsen four decades after he had seen it at the end of the war. During his visit he said the following, I do not bring forgiveness, I do not bring forgetting only the dead have the right to forgive. Only the dead have the right to forget. The living have no right to forget. This is what he said in his testament, I do not carry with me in my heart. 35 years have passed since then, and a couple of hours from now, I will end my state visit in the camp of Bergen Belsen. And I would like to repeat his words before you as a representative of the German people from all political parties. I declare that Jewish people does not forget, not just because of our obligation towards past generations, also from our obligation towards generations of the future. It is not easy to actually do this remembering because it is shattering, it is complex, it is hard, it is painful, it is painful for Germans because there will always be people who will deny everything, who feel as if they were accused who just want to see the past, the past, later rest. And in a very different way, it is also very complex and painful in Israel too. Remembering the Shoah is an indispensable part of our national identity. A people that holds these deepest abysses, these horrible experiences in its collective memory, that is not a people like any other. But despite all that, even if we cannot meet in our past of remembrance, we still have to meet. We have to give meaning to remembrance. Which means remembering is an obligation, it's a moral obligation, it is a responsibility. You cannot bridge the past, you cannot do away with the past, but the future, the future is in our hands, and it holds great responsibility. So it belongs to both of us, it belongs to us, it belongs to you, because only together can we give meaning to remembrance. Only by meeting, only by deepening relations, only by holding up this mutual obligation towards freedom, humanity, democracy, apart from our obligation to freedom, security, and the prosperity of the Jewish people. Both our peoples will give meaning to remembrance and it will be used as an example for the entire mankind. Ladies and gentlemen, history links the Jewish people and the state of Israel with the German people. It is a link of memory, of meaning, which takes place in the past, in the present, but also in the future. If we look at the past, we have to still discuss research, the Shoah. We have to look at what happened in a realistic way. We have to fight against everybody who denies it. We have to turn every stone in order to try and understand what cannot be understood. If we look at the present, the main task that obligates us every day to not ignore hate, be it on the web, be it in social networks, be it in the streets, be it in the political centers. We have to fight anti-Semitism and racism wherever we encounter it, decisively, without any compromise. We have to seek the way that we want to walk together. We have to seek the way that we walk together, oriented towards freedom, justice, mutual respect, tolerance and partnership. If we look at the future, ladies and gentlemen, I would say that this obligation towards the future is a privilege. The partnership between Germany and Israel is known all over the world. It is our task to use this in order to create a glorious future, not just for our countries but for the entire world. The State of Israel is an example of reconstruction of success, of overwhelming contribution for humanity in culture, in medicine, in academics, in science and many others. Throughout the years, Israel has co-operated and established partnerships with many European countries and many sectors. We participate in efforts to solve crises worldwide, among others, the climate crisis. My country plays an important role when it comes to finding ways how to deal with the climate crisis to develop new technologies. In the Middle East, Israel is the engine when it comes to established partnerships in the region. We have new peace agreements. We have normalization. We have dialogue. We have rapprochement. And in this, we have created good, neighborly relations and growth in our region that we've never seen before. We never shied away from criticism. We never ignored criticism or tried to suppress criticism. But one thing we will demand from our critics to check everything and to make sure that the criticism is embedded in truth. We extend our hand to everybody, to our neighbors and deepening our relationship with our neighbors. That is a real thing. And that refers to the entire region. That is the truth, too. Israel has always and will always strive to establish neighborly relations with all states and peoples of the Middle East. And this effort will continue. And it is, of course, also true for our Palestinian neighbors. This is part of the hope and the objective that we will never give up, because this obligates us and Palestinians to face reality, to undertake every efforts to make this future a good future. Our Palestinian neighbors have to fight terror and have to seize terror right away. Just yesterday, President Steinmeier and myself spoke to relatives and leader representatives of Germany and Israel. And when we commemorated the 50th anniversary of the attack of the Munich Olympic Games where 11 Israeli athletes were killed, who wanted to participate in these games, the Palestinian terror organizations are responsible for these horrific killings, a crime that not just hit the very principles and foundations of human dignity, but also hit the entire spirit of the sports and the spirit of Olympic Games. And the same holds true for other similar horrific terrorist attacks that still are being committed. And for that, President Steinmeier, Chancellor Scholz, I would like to thank you as well as I'd like to thank the Federal Republic and the government of the State of Bavaria who enabled this exchange with the victims' relatives. What is embedded is a historic investigation of what happened to bring out the truth in order to alleviate this pain that the people affected will always hold. Thank you for this very moving ceremony that took place in Munich with the Olympic Associations, together with young people from Israel and Germany, together with the families of the victims. Ladies and gentlemen, Israel is part of the international efforts to keep radical forces at bay who spread terror and mistrust and who threaten every citizen in this world. During our times, dark, hateful forces led by Iran do not only threaten the State of Israel in the Middle East, but the entire world order. And from this place here in Berlin, this important place in Berlin, I call upon the nations of the world to decisively oppose Iran and its plan to develop nuclear weapons. Weapons of mass destruction in the hands of a member state of the United Nations that every day calls for the annihilation of another member state of the United Nations. That is unacceptable. Threats and actions to destroy Israel are not acceptable. And let me say this quite clear. Whoever denies the Shoah, who acts from hatred and aggression, who denies the right of existence of the State of Israel has no right to claim to be party to any negotiations, does not have the right to claim any financial support or any other form of support. The international community has to stand on the right side of history. They have to formulate clear preconditions, have to decide on clear sanctions. They have to build a wall between Iran and nuclear capabilities. Stand with resolve. Do not give up. The State of Israel will defend itself with all means. And I call upon the entire world to not just stand, I live by. Ladies and gentlemen, dear representatives of the German people, Germany and the German people have developed and rebuilt the entire nation after the Second World War in a very admirable way. Germany is a known country. It is a responsible, it's a leading country in many areas of the world. It's the initiator of a never-seen uprising, economic upswing here in the entire world. Germany is a leading force in Europe, in NATO. And it is now that this becomes particularly invisible when it comes to supporting the Ukraine and the inviolability of Ukraine. Germany will play a leading role when it comes to finding a solution to this terrible tragedy. All of us who wanted to see the suffering end it as soon as possible, we want to see freedom and stability of return to Ukraine and in all of Europe. What Germany achieved in the social field, equal rights, taking care of the weaker parts of society, erecting infrastructure, all that comes together as a picture of humanity and the dignity of mankind and this can serve as an example to all of Europe. And it is held in high esteem throughout the world. The esteem that Germany enjoys throughout the world and makes it a leading nation in the world is based on one thing, is based on the obligation to stand up to its past but also an obligation towards the future of mankind. Germany has proven and continues to prove that it takes this culture of remembrance seriously. At the same time, it puts major efforts in innovation, in entrepreneurship, in science and technology, in research and many other fields. All areas on which our future depends. The State of Israel is proud to be a partner of the State of Germany and we cherish the friendship of this deep alliance between our two countries. The German contribution towards security and the stability of Israel is held in high esteem in our country. I'm convinced that our joint value is the deep friendship between our two countries would continue to foster this relationship, this partnership so that we continue along this path that Germany and Israel walk together side by side hand in hand towards a prosperous future. Ladies and gentlemen, the Israeli poet I quoted at the beginning of my speech, my speech, my people who live, those are the components of his name, again refers to the stone on his desk in Jerusalem, which is part of a gravestone in the Jewish cemetery. At the end of my speech, I would like to again quote from a poem and I want to express my hope that these words can be the foundation for our partnership. Let us nourish by the past and the future. These are his words. This stone on my desk gives me peace. It is a stone of truth, immovable, wiser than any stone of the wise. It is a stone, a fracture of a gravestone. It is a stone of witness of everything that ever happened and of everything that will ever be. It is a stone of amen and a stone of love. Amen, amen. This is how it should be. Thank you. President Herzog there in his first official visit as president of the State of Israel


And welcome to the NPTEL course on an Introduction to Programming through C++. I am Abhiram Ranade. The lecture of today is on Virahanka numbers. Virahanka was an ancient Indian prosodist from the 6th or 8th century. And prosodists are people who study patterns of rhythm and sound in poetry. Virahanka asked a question about poetic meters and interestingly enough solved it using recursion. So that is what we are going to study today. So let me tell you a little bit about what a poetic meter is. A poetic meter is characterized by the number of syllables in the meter and the duration of each syllable. So syllables have either a short duration or a long duration and the long duration is just twice the short duration. So for example, a poetic meter could be described by the sequence of characters S L S S L S. So that means that there are 6 syllables and the total duration is 8. The additional 2 counts come from the 2 long durations. So let me give you an example for a poetic meter, an actual poetic meter. And here is a poetic meter called Shardul Vikridith. So this is used in many many poems and many many shlokas. And perhaps here is a familiar shloka that is in this meter. So I am going to sing the shloka and you will note that some of the syllables are going to be short and some of the syllables are going to be long. And it is the long and short, the long and shortness of the syllables which gives it, gives this shloka a certain character and that is what a poetic meter does. So ya kundendu tushara hara dhavala ya shubh bravastra vruta. So as you can see the first ya is a long syllables, kund is a long syllables, den is a long syllables. But then you sing ya kundendutu, so dutu are short syllables. So overall there are 19 syllables and the total duration is 30 if you count it. So prosodists were concerned with such poetic meters and they asked questions such as how many different poetic meters can there be and they sometimes even wanted to enumerate the poetic meters. So Virahanka's question was how many poetic meters exist of total duration d?  So if d is equal to 1, so you want a poetic meter of duration 1, well you do not have a choice. The only poetic meter possible of duration 1 is sort of a very trivial thing, a poetic meter which just has one short syllable. If you go to duration 2, then the duration 2 can be made either by one long syllable or it can be made by two short syllables. So at this point you have two possible meters, 3 if you work this out, if you try this out and if you try to do this exhaustively what do you get? Well you get that duration 3 can be made up by three short syllables or a short and long or a long at the short. So there are three poetic meters of total duration d. And for d equal to 4, you may again work this out. So there might be all four shorts or you might have SSL, SLS, LSS or just two longs. Now we want to ask this question in general and so let us say that v of d denotes the number of poetic meters of duration d. And what we have worked out is that v of 1 is 1, v of 2 is 2, v of 3 is 3 and v of 4 is 5. Virahanka wondered whether there is an easy way to calculate v of d. So that was his question and we are interested in this not because this is a course on poetry or prosody or anything like that but because this question has some interesting things to say about recursion. So let me tell you Virahanka's solution. So Virahanka observed that the first syllable of every meter must be S or L. Of course I mean there is nothing much over here but that is just the first step. So then he said that look let me use S of d to denote the set of meters of duration d in which the first syllable is an S. So let us take an example just to make sure that this notation is clear. So if you take d equal to 4 then the set of all meters of duration 4 which we just worked out is this and of these the first 3 have the first syllable S. So S of 4 is just those first 3 meters. And L of d we are going to use to denote the set of meters of duration d with first syllable L. So clearly L of 4 are the remaining so these 3 belong to S of 4 these 2 are the remaining ones from all the set of all meters of duration 4 and they clearly belong in L of 4. And clearly V of d the total number of meters of duration d must be equal to the short meters plus the long meters and since S of d and L of d are sets we are just taking the sizes of those sets. So checking it for d equal to 4 what we have is V of 4 is 5 which we counted over here and 3 of these comes from the short syllable first and 2 of these come from the set which has the long syllable first. So I am just using d equal to 4 just to make sure that our notation is being well understood. Now here is the key question to ask which Virahanka did. So suppose I remove the first letter from every meter in S of d what happens? What is what remains in S of d? If you remove the S from anything the duration will decrease by 1. So you started off with meters of length d so if you remove the first meter which is going to be S in the set S of d then you will be left with meters that have duration d minus 1. And interestingly enough not only will you be left with meters which have duration d minus 1 but in fact you are guaranteed to have all meters of duration d minus 1 after you remove the first syllable from the meters in S of d. So again let us go to our running example. So we have S4 which is this and let us remove the first syllable from these. So what do we get? Well from this 4 SS we get 3S SSL gives us SL and SLS gives us LS. So indeed if you look at the durations of these meters they are 3 and if you look back to what we did earlier these are exactly all the meters of duration 3. Now we should probably prove this so here is a very simple proof. So I am going to do it with respect to this set but you can see that it is really a general argument. So suppose this was not the entire set of meters with duration 3 and maybe there was some additional member that we did not have in this set. So suppose I take that additional member which has to be different from this and I add an S in front of it what do I get? So I will get a meter which starts with S and has total duration 4 but then that would have to be present over here. So either but that clearly is not the case so that means this is a set which contains all meters of duration 3. And since the number of all meters of duration 3 is V of 3 in general I can say the size of the set S of D must be V of D minus 1 because after I remove one letter from all the meters the number of meters in this set does not change. So that continues to be the cardinality of S of D and now we have established that is the same as the number of meters of size D minus 1 and therefore that must be equal to V D minus 1. So we can do the same thing removing the first letter from all meters in L of D. So what do we get? Well if you remove the first letter from L of D, L has duration 2 so now everything that remains will have duration D minus 2. So in fact not only will the meters that remain have duration D minus 2 but in fact by the same argument all meters of duration D minus 2 will be present. So again let us check this if I look at this L4 if I remove the first L what do I get? I get this and indeed these are all meters of duration 2 there are only 2 meters of duration 2 and these are here. So again the number the size of L of D must be equal to V of D minus 2. So combining this, this and this what do we get? Well V of D is cardinality of S of D but that is V D minus 1 plus L of D that is V D minus 2. So this is what we get and I should observe that this is valid only for D greater than 2 because if D is not greater than 2 then V of D minus 2 would not be bigger than 1 and our formulae all our discussion is for having at least one syllable in the meters that we are talking about. So our discussion is really about, so in our discussion whatever sets we are talking about should have say V of D so this has to be bigger than 0. So D had better be bigger than 2. So what have we discussed at this point? Well, so we have introduced Vran K's problem and that was find the number V D of poetic meters having duration D and we can work out by hand that V of 1 and V of 2 equals 2 and we said that working out larger V of D by hand is tedious and error prone and so we derived a relationship which is that V of D equals V of D minus 1 plus V of D minus 2 for all D greater than 2. Next we are going to write a program to calculate V of D but let us take a short break.


Here's deficit. The market's down. It's a sell-off. Packman is at the market. Gobbling it up. Don't worry, we're not following him. East Timor lives.