Book-stacking Problem

Book-stacking Problem 📄𝗙𝗶𝗹𝗲 𝗟𝗶𝗻𝗸 https://drive.google.com/file/d/1qxlrL5wuJu-Xy0UnBPlLMqNMQ7ZOeBUB/view?usp=sharing 📫𝐎𝐮𝐫 𝐅𝐁 𝐏𝐚𝐠𝐞: https://www.facebook.com/ScienceWorld-106933907791981 📚𝐃𝐚𝐯𝐢𝐝'𝐬 𝐁𝐨𝐨𝐤𝐬 📕 𝗪𝗲𝗶𝗿𝗱 𝗠𝗮𝘁𝗵𝘀: 𝗔𝘁 𝘁𝗵𝗲 𝗘𝗱𝗴𝗲 𝗼𝗳 𝗜𝗻𝗳𝗶𝗻𝗶𝘁𝘆 𝗮𝗻𝗱 𝗕𝗲𝘆𝗼𝗻𝗱 (https://www.amazon.com/Weird-Maths-Agnijo-Banerjee-Darling/dp/1786072645) 📙 𝗪𝗲𝗶𝗿𝗱𝗲𝗿 𝗠𝗮𝘁𝗵𝘀: 𝗔𝘁 𝘁𝗵𝗲 𝗘𝗱𝗴𝗲 𝗼𝗳 𝘁𝗵𝗲 𝗣𝗼𝘀𝘀𝗶𝗯𝗹𝗲 (https://www.amazon.com/Weirder-Maths-At-Edge-Possible/dp/1786075083/) 📗 𝗪𝗲𝗶𝗿𝗱𝗲𝘀𝘁 𝗠𝗮𝘁𝗵𝘀: 𝗔𝘁 𝘁𝗵𝗲 𝗙𝗿𝗼𝗻𝘁𝗶𝗲𝗿𝘀 𝗼𝗳 𝗥𝗲𝗮𝘀𝗼𝗻 (https://www.amazon.com/Weirdest-Maths-David-Darling/dp/1786078058/) ** The kindle versions are available *** For more details : http://weirdmaths.com/ 📄𝗧𝗿𝗮𝗻𝘀𝗰𝗿𝗶𝗽𝘁𝗶𝗼𝗻: How much of an overhang can you achieve by stacking books on a table? Let’s assume that each book is one unit long. To balance one book on a table, the center of gravity of the book must obviously be somewhere over the table. To achieve the maximum overhang of a stack of books, the center of gravity of the stack should lie directly above the table's edge. The maximum overhang with one book is obviously 1/2 unit. For two books, the center of gravity of the first book should be directly over the edge of the second, and the center of gravity of the stack of the two books should be directly above the edge of the table. The center of gravity of a stack of two books is at the midpoint of the books' overlap, or (1 + 1/2)/2, which is 3/4 unit from the far end of the top book. It turns out that the overhangs are related to what’s known as the harmonic sequence. If there are n books, the maximum overhang is 1 + 1/2 + 1/3 + ¼ all the way up to 1/n, all divided by 2. With four books, the overhang (1 + 1/2 + 1/3 + 1/4)/2. And because this is greater than 1, no part of the top book is directly over the table. If we want an overhang of 2 book-lengths it turns out that we’d need 31 books. For an overhang of 3 book-lengths the number of required books jumps to 227. And if you were ambitious enough to go for a six book-length overhang, you’d need 91,380 books stacked in the optimal way to achieve your balancing act. #book_stacking #block_stacking #problem