Latest news with #Bhaskara
Hindustan Times
4 days ago
- Science
- Hindustan Times
Chicken and egg — and duck too
Here at Problematics we usually aim for puzzles that are not the kind you would find in a textbook, but there are exceptions. Some puzzles that can be solved with textbook methods are still interesting because of the way they are packaged or because of their pedigree, with illustrious minds having dwelt on them at some point in history. A prime example of puzzles that are delightful because of both packaging and pedigree are the problems in Bhaskara's Lilavati. While those are widely known, I recently found one that I hadn't come across earlier. It is said to have appeared in a book by the great Euler, and described by the French writer Stendhal before making its way into the writings of the late Russian mathematician Yakov Perelman. To insulate the solution from an internet search, I have added my customary modifications to the version described by Perelman. I have changed the currency to Indian rupees, and tinkered with the prices to bring them within a range that is credible for the story into which I have packaged my adaptation. The story, of course, is entirely my own. #Puzzle 145.1 A family of poultry farmers collects 100 eggs one morning. They are all chicken and duck eggs, the distribution being unequal. Handing the chicken eggs to their son and the duck eggs to their daughter, the farmer parents send them off to the market. The price for each kind is fixed, with the duck eggs being costlier than chicken eggs, as is the case in most places. Each child sells his or her full share of eggs at the respective fixed rates. In the evening, when they compare their earnings, they are thrilled to find that both have made exactly the same amount. I am no farmer, but the internet tells me that hens and ducks lay about one egg daily at the peak of their productive years. It is not surprising, therefore, that the same birds at our farm lay the same number of eggs the following morning. In other words, the family has 100 eggs again, and the unequal distribution of chicken and duck eggs is the same as on the previous day. Mother segregates the produce into a number of baskets, the chicken eggs on one side, the duck eggs on another. Father passes the orders: 'Pick up your respective shares and come back with the same earnings as you did yesterday.' The kids get mixed up, of course (how else would there be a puzzle?) The son picks up the duck eggs by mistake, and the daughter takes the rest. Neither of them notices that his or her count is not the same as on the previous day. At the market, the boy sells the duck eggs at the price for chicken eggs, and his sister sells the chicken eggs at the price for duck eggs. When they compare their earnings in the evening, the boy is alarmed. 'I got only ₹280 today. I don't know how I can explain this to Father,' he says. The girl is equally puzzled about her collection, but pleasantly so. 'I don't know how, but my earnings rose to ₹630 today,' she tells her brother. #Puzzle 145.2 MAILBOX: LAST WEEK'S SOLVERS Hi Kabir, Assuming that the store owner initially bought cat food for 31 cats for N days, or 31N cans. As each cat consumes 1 can/day, the total consumption reduces by 1 can every day. Again, all cans were consumed in one day less than twice the number of days originally planned, or (2N – 1) days. Thus the total number of cans is the sum of an arithmetic progression of (2N – 1) terms starting 31, and with a common difference of –1. The sum of the AP is: [(2N – 1)/2][2*31 + (2N – 1 – 1) (–1)] = 65N – 32 – 2N² Equating the above to 31N and simplifying, we get the equation 2N² – 34N + 32 = 0. The roots of this equation are N = 16 and 1. As 1 day is not viable, N must be 16. So the total number of cans bought initially = 31*16 = 496. And as it took (2N –1) = 31 days to finish the whole stock of food, only 1 cat was left unsold. — Anil Khanna, Ghaziabad *** Hi Kabir, Suppose the cat food was initially ordered for N days. Then, the number of cans ordered = 31N. Also, suppose K is the number of cats remaining unsold when the food stock got exhausted. On any day, the number of cans consumed is the equal to total number of unsold cats. Thus the total cans consumed = 31 + 30 + 29… + (K + 2) + (K + 1) + K = (31 + K)(31 – K + 1)/2 i.e. 31N = (31 + K)(31 – K + 1)/2 For the right-hand side to be a multiple of 31, K has to be 1. This means 31N = 32*31/2, or N = 16. The number of cans = 31 x 16 = 496. The food lasted for 31 days. If we add one more day, we get 32 days which is twice the original period of 16 days. — Professor Anshul Kumar, Delhi From Professor Kumar's approach, it emerges that the puzzle can be solved even without the information about the cans being exhausted in (2N – 1) days. Many readers, however, have used this bit in solving the puzzle. Puzzle #144.2 Hi Kabir, The puzzle about the party trick is fairly simple — you randomly tap on any two animal names for the first and second taps and then tap in the order of length of the animal names — i.e. COW (third tap), LION, HORSE, MONKEY, OSTRICH, ELEPHANT, BUTTERFLY AND RHINOCEROS. Obviously, this trick will get old very soon because your tapping pattern will become predictable to a keen observer. — Abhinav Mital, Singapore Solved both puzzles: Anil Khanna (Ghaziabad), Professor Anshul Kumar (Delhi), Abhinav Mital (Singapore), Kanwarjit Singh (Chief Commissioner of Income-tax, retd), Dr Sunita Gupta (Delhi), Yadvendra Somra (Sonipat), Shishir Gupta (Indore), Ajay Ashok (Delhi), YK Munjal (Delhi), Sampath Kumar V (Coimbatore) Solved #Puzzle 144.1: Vinod Mahajan (Delhi)
New Indian Express
25-04-2025
- Science
- New Indian Express
Goldfinger who was a hero: ISRO saw success after success with Kasturirangan at the helm
O Chairman! My Chairman! You could tweak Walt Whitman's stirring lament a tad and encapsulate why the makes this a truly sad Friday for me. The former chairman of ISRO was more than a friend to me, at all times. There was a saying within ISRO that Dr Kasturirangan was a person with golden fingers. We have lost that golden touch and much more. The first time I met Dr Kasturirangan was at the Physical Research Laboratory, Ahmedabad, in 1967, immediately after I joined the Thumba Equatorial Rocket Launching Station. He was a research scholar at the Physical Research Laboratory then, and we were staying together at a hostel there for several weeks. That was how our association began. When he took over as chairman of ISRO in 1994 after UR Rao's departure, I served under him for all the nine years of his tenure. This was in various capacities -- as Director of Vikram Sarabhai Space Centre, Liquid Propulsion Systems Centre, as Project Director for PSLV, etc. We were therefore intimately connected professionally, and the mutual discussions and interactions I had with him were highly beneficial. He rose to the helm of ISRO just after the failure of the first Polar Satellite Launch Vehicle in 1993. It was a herculean task to recover from that and ensure a successful mission as quickly as possible. But then again this was Dr Kasturirangan we are talking about. Under his leadership, in 1994 itself, we were able to achieve the first successful launch of the Polar Satellite Launch Vehicle. And since then, we have never had to look back. Almost all the PSLV launches were successful. Whatever he touched, was a success. After completing his PhD, he initially joined the ISRO Satellite Centre at Bangalore, working on projects like Aryabhata, our first-ever satellite, and Bhaskara. Later, he was a key figure in developing the first Indian Remote Sensing satellite, IRS-1A, which was launched from Russia. Eventually, this IRS series was launched, using PSLV. There were dozens of successful missions, both launch vehicle and satellites, during the nine years under his leadership. As a leader, he focused on motivating team members, and providing the right kind of technical guidance. This led to success after success. That is how he came to be known as the chairman with golden fingers. In earlier stages, during Satish Dhawan's time, it was more about infrastructure building and technology initiation. During UR Rao's time, it was the ordeal of technology development and demonstration. Without any foreign help, developing sophisticated technologies for launch vehicles or satellites was a huge challenge. During this process, we encountered a few failures. But ISRO had developed the habit of learning from these failures and correcting mistakes to avoid repetition. This culture really helped us. Looking back, the foundation built by Professor Dhawan and Professor Rao was crucial in building the future development course of ISRO. Dr Kasturirangan truly benefited from this legacy. Remarkable handling of the infamous spy case He was heading the organisation when in 1994. The way he handled the situation was remarkable. Without getting into the limelight, he silently managed the situation. First, he ensured that there was no leakage of any technical information from ISRO to outside agencies. Then he worked with the Prime Minister at the time, Narasimha Rao, and brought in the CBI to look into all aspects and quash the false allegations raised against ISRO scientists. Not only that, he ensured there was no demoralisation in the organisation, which can often happen during such crises. As a result, the third PSLV launch was made within a year, after the so-called spy scandal. He also played a major role in the management of the cryogenic contract with Russia. Initially, it was supposed to involve technology transfer but due to the geopolitical situation and the breakup of the Soviet Union, it was denied. This led to the responsibility of developing cryogenic technology in the country falling upon ISRO. Under his leadership, ISRO took the initiative of establishing various test facilities for it and developed the first indigenous cryogenic engine. Gentle giant who believed in science helping the common man One of his significant contributions was the application of satellite technology for the benefit of the common man. Earth observation satellite data was widely used for agriculture, forestry, water resources management, fisheries development, and assessing damage during severe events like cyclones or heavy rainfall. This helped the nation implement several practical projects providing advance warnings about weather events, managing water resources, and even forecasting agricultural yields, using remote sensing. He believed in using satellite technology for development and implemented many programmes, in cooperation with various central and state government departments. As a person, he was very gentle, soft-spoken, and analytical. In case of an issue, he would go through it himself in fine detail to ensure no mistakes were made at the design or implementation stages. When it came to fund utilisation, he was extremely careful and made sure there was no wasteful expenditure in the organisation. He was always there with the working teams at ISRO's various labs across the country, motivating people, listening to their problems, ensuring relief for those in need, and encouraging those who were sometimes depressed by failures during the developmental stages. Post-ISRO years: Big NEP contribution and Western Ghats After leaving ISRO, he played a significant role in the Planning Commission as a member, and also served one term as a Member of the Rajya Sabha. After his tenure as a parliamentarian, Dr Kasturirangan played a very major role in giving future direction to the education system in the country, as the chairman of the NEP commission. This report is now being taken up by the Centre for implementation. It stresses the need for identifying talents at a young age itself and laying focus on developing the talents of each individual, as per their abilities. It also provides a flexible scheme for continuing higher education so that the financial burden of higher education can be managed effectively. This is a unique scheme and it's going to revolutionize the entire education system in the country. As far as Kerala is concerned, what is still remembered is the Western Ghats Environmental Impact Assessment Report. Earlier, Dr Gadgil had submitted a detailed assessment about the resources in the Western Ghats and how they had to be protected. But there were many practical issues. Dr Kasturirangan carefully analysed the situation using satellite images, interacted with various stakeholders and brought out the Kasturirangan Report, parts of which have been implemented to safeguard our prestigious natural resources in the Western Ghats. He was a great scientist, a highly practical individual, a social scientist and above all, a very fine human being. It is a great loss for the nation. (The author is former Chairman, ISRO)
