Combinatorics formulas. Combinatorial problems with placement constraints. Counting the number of placements

The topic that the author explores is books and bookshelves. He asks the question: is the modern state of affairs, when books stand vertically on horizontal shelves, so obvious and inevitable? Readers will follow how a scroll from Antiquity turns into a codex, and that, in turn, into a book to which we are accustomed, and will learn how the problem of storing book collections was solved at different times. This is a lavishly illustrated and engagingly written book about the book - how it came into being and how we learned to preserve it.

I. Books on bookshelves

The chair in which I am accustomed to read faces the bookshelves; I see them every time I take my eyes off the page. Of course, “I see them” is only a figure of speech: how often do we really see what we look at day after day? In fact, I see the books rather than the shelves themselves. If I specifically start thinking about them, focus my gaze differently (as if I were looking at optical illusions in which the steps of a staircase go up and down or a cube turns right and left), then I will see the shelves, but rather just their edge or the bottom of the upper shelves; It is much less common to see entire shelves. Even when they are empty, I pay attention not to them, but to the absence of books, because the meaning of the shelves is their purpose.

To tell the truth, I can’t even see books without shelves. The bottom edges of the books rest against the shelf, their rows standing straight, defying the force of gravity. The upper edges form an uneven line, but this is also due to the shelf on which the books stand, and the straight line of the shelf only emphasizes this unevenness. Books and bookshelves are a technological system, and each element affects how we see other elements. Since we deal with both books and shelves, we ourselves become part of this system. Thanks to this, we look at it and its components differently and interact with them differently. This is the nature of technology and the artifacts it creates.

Looking at bookshelves is not such an easy task. In my office, the shelves go from floor to ceiling, taking up almost an entire wall, but my office is small, so I can’t move far enough away to take in that entire wall at once. I couldn’t do this even when I just started using this cabinet and the shelves were empty. It doesn't matter where I stand in front of the shelves: I see the undersides of some shelves, the tops of others, the left sides of some vertical supports and the right sides of others. I can't see even one shelf in its entirety. Of course, it would be easier to think that all the shelves are the same and, having seen the bottom of one shelf, you see the bottom of all the others, but in such, albeit widespread, philosophizing, something still does not suit us.

One late evening, as I was sitting in a chair and reading, for some unknown reason I suddenly looked differently at a bookshelf filled with books. It struck me as a functional item that is taken for granted or not noticed at all; the shelf reminded me of a bridge under traffic, and I wanted to know more about the nature and origin of this ubiquitous object. But where to start? Maybe you should ask yourself why the shelf is horizontal, but the books on it are vertical? Or is it so obvious that it requires no explanation? Perhaps we should go the other way and ask why we put books with the spines facing out? Or is this the only logical way to put them? Maybe books and shelves, like nuts and bolts, can only be combined one way?

It turns out that the history of the bookshelf is inextricably linked with the history of the book, and vice versa. Of course, books could exist without shelves. You can imagine how in the Library of Congress or even in a district library, books are stored in boxes, lying in piles on the floor - like firewood or coal. But there would be no bookshelves if there were no books. This does not mean that there would be no shelves at all, but they certainly would not be bookshelves. The bookshelf, like the book, has become an integral part of civilization. If there is a bookshelf in the house, this is an indicator that the owner is a civilized, educated, cultured person. The presence of bookshelves has a serious impact on our behavior.

Why are book authors often photographed in front of bookshelves? After all, they didn’t write the books that are in the background! Maybe they want to show us how many books they read to write theirs, and that we won’t have to read them if we delve into the detailed study or historical novel they wrote, which has detailed notes and an extensive bibliography. It is unlikely that the book with such a photo printed on the cover is itself on this shelf - perhaps this is our way of hinting that we need to go to the bookstore and buy this book so that there is a complete set on the shelves.

But is it really possible to stock the shelves completely? In America alone, more than fifty thousand books are published every year. Can a person read that much in a lifetime? It's easy to calculate. Let's say we read about one book a day. This means that every three years we read about a thousand books. Let's say we read the first book at the age of four and lived a long life - up to ninety-four years. It turns out that we have read about thirty thousand books in our lives. How much space is needed for this number of volumes? Let's assume that each book takes up 2.5 centimeters on the shelf. This means that the total length of the shelves should be about 762 meters. This number of books would fit in a house with six or seven large rooms, where each wall is occupied by shelves. This is no longer a house, but a bookstore, or a public library in a small town.

But if we go into such a house, what will we see there: books or bookshelves? What do we see when we enter the library? Almost always, only books attract our attention. The shelves, like the steps of a staircase on which people stand taking photographs, remain unnoticed: they are there, but they seem to be absent. They play a supporting role. But at the same time, the absence of shelves is striking. If we see that there is not a single book, not a single bookshelf in the house, then we will think about the owners like this: do they really just watch TV?

It's funny that the bookshelf is a popular television prop: it appears in the background every now and then in various television interviews - in a variety of shows, from Today to Nightline (1) . On the CSPEN TV channel (2), congressmen and senators hold press conferences in front of a bookcase that just fits into the frame (I wonder if the books in it are real?). When Newt Gingrich (3) spoke in a tie with a picture of bookshelves, you could say that he had books in both front and back. Often, against the backdrop of bookshelves, journalists interview lawyers and professors - probably the producers’ idea is that the authority of the invited experts is supported by the authority of the books.

A bookshelf serves as a support for books, but it also needs support. She is not only a decoration, but also a stage on which the books are lined up to receive applause. But although the role of the bookshelf in the history of civilization is undoubtedly important, it is rarely mentioned in the program of this performance: the shelf is an extra in it, it is taken for granted, and is simply ignored. There are many anecdotal examples of this.

One day, when we had guests, my colleague's wife went into my office to feed her newborn baby. Returning some time later with a sleeping child in her arms, she said to me: “I hope you don’t mind that I ran through your bookshelves - I found several books there that I was pleased to remember.” Of course, there is nothing strange in the fact that she did not say a word about the shelves themselves. But when another guest came into my office on another occasion, he looked at the books with such attention and did not notice the shelves at all that it is worth telling about this in more detail.

One fine spring day, this guest found himself in my office: I was looking for a book to give him to read on the plane. Soon he began not just to look at books, but to carefully leaf through them; he studied books with a single-mindedness that was familiar to me. Looking through other people's books is a game of chance, or even an act of voyeurism or an exercise in home-grown psychology. My guest didn't seem to miss a single volume, and he told me that he always found it interesting what books people bought and read. This interest is understandable: my guest was a psychologist, a specialist in cognitive research, who worked as a consultant on computer interfaces. At the time, he was advising a large office equipment company on what products to develop and improve on. He is the author of thoughtful works on the design of everyday objects, with special attention to the use of these objects. I have read his books and I think that he is incapable of losing sight of anything, no matter what he looks at.

On the morning of the same day I showed him the city. We stopped by the new building for the Faculty of Political Studies, which has received much praise for the care with which the architect has given its future use. As soon as we walked inside, it immediately became clear that this was an unusual building. Numerous offices and conference rooms open onto galleries that line two sides of the common room, while the other two sides are continued by tiers of open spaces that also open onto and frame the atrium. When going from one part of the building to another, you will definitely pass along a gallery or staircase from which this common room is visible; In such a building, visitors must meet each other by chance every now and then - no doubt this is how it was intended. The layout reminded me of the National Humanities Center, where visitors pass through a common room that doubles as a dining room where visiting scholars gather to socialize, working on books on everything from pencils to phenomenology. My guest was very impressed with the care that had gone into the design of the new building: he paid attention to details that most of us would not have noticed, such as the light fixtures above the notice boards and the door fittings, which he wrote about with special understanding and understanding. love. I was already planning this book at that time, so I was hoping to see how bookshelves were arranged in the offices of the new building. Alas, we came on a Saturday and all the offices were closed.

Back in my office, we talked not about objects, not even about books as objects, but about the ideas they contain, and how different categories of books are grouped on my shelves. My guest found familiar books in my room that he, of course, expected to see - for example, “The Soul of a New Machine” by Tracy Kidder (6) and many books about bridge construction, but some books surprised him. I explained that, for example, books about the design of computer programs had been sent to me and given to me by readers of my own books about the design of bridges and other useful objects. Since I am of the view that design is design, regardless of its application, my collection of books reflects this unity, if not my over-indulgence in certain ideas. But I confided to my guest that it can be difficult for me to decide where to place a book that deals with several aspects of a single topic. My guest probably had an opinion about how I read and work in my office, but then we started talking about computers and what characteristics of a laptop I should pay attention to (I had just told the guest that I was going to buy one).

If my guest formed an opinion of me by looking at the books on my shelves, this confirms one of my hypotheses: even the most attentive people who look at useful objects do not notice the infrastructure that maintains these objects. My guest said nothing about the shelves themselves, although I tried to lead the conversation in that direction. He couldn't reach the very top shelves, but even this did not cause any reaction from the inveterate critic of everything in the world - from the design of telephone systems to the location of switches. The "dust and silence of the top shelf" of which Lord Macaulay wrote was also not discussed. Once installed and loaded with books, a shelf has no moving parts or any other purpose than to stay in place and support a row of books. It is similar to an ordinary bridge in the countryside: for those who walk on it every day, it is there, but it is not there. However, if the bridge is suddenly carried away by a flood, all the people in the area will talk about is it. This is generally characteristic of technology: its presence is in its absence.

As I began working on this book, I began to see shelves where I had previously only seen books, but not everyone shared my vision. Having found myself at a dinner with a historian who himself had built wall-length shelving - just the kind that would be convenient for storing paperback books, historians especially have a lot of them - I started talking about the bookshelves that I used to use when I was visiting him, I didn’t pay attention. At first we talked about the pride of the master (in fact, it is not easy to build such a rack), and then, of course, we switched to more general topics - books and their arrangement on shelves. I was thinking a lot at the time about how books were stored in the Middle Ages and the evolution of bookshelves, so after dinner I tried to start talking about shelves again. I was interested to learn that the origins of these objects are not very well known even to historians, especially those who do not specialize in the Middle Ages. A few months later, in a conversation with a retired English professor, I was again convinced that scholars of later periods do not always understand the physical nature of medieval books, and do not always know that they were chained to the shelves.

I heard not only from scientists, but also from librarians that few people know the history of the book and its care, as well as the evolution of furniture for storing and displaying books. I have repeatedly turned to one rather old work - “The Chained Library” by Burnett Hillman Streeter. This very title aroused the curiosity of the librarians to whom I approached for this book. It was published in 1931 (7), and, apparently, for the first ten years after that, readers asked for it regularly, although infrequently. But the last return date stamped on the library card is October 28, 1941. Judging by the signatures on the form, which still lies in a pocket on the back flyleaf, no more than ten people read this book in one of the best research libraries in the country. At least I couldn't find any evidence that it was ordered at least once in the next ten years. I cannot trace what her fate was next, because in the early fifties of the twentieth century, accounting procedures in libraries changed. Since then, the form with return marks has remained on the back flyleaf; This is a sign of the time when the librarian probably knew everyone who signed the form by sight. One way or another, I realized that what the author of “The Chained Library” writes about (for example, the accounting of book circulation that existed before) is, as a rule, unfamiliar to young librarians. They did not share my interest in the history of libraries—at least not my interest in library furniture and methods of storing books.

After reading The Chained Library, and before it John Willis Clark's seminal work on the subject, The Care of Books, I went to Yale University's Beinecke Library, home to one of the world's finest collections of rare books. This library was shown to me by a knowledgeable and sympathetic person, but when asked whether there were books in the library that had traces of the chains that once chained them to the shelves, he was unable to answer. However, the library employee searched the computer catalog for the word “chain.” Many of the findings involved the chain stitch used to sew together old bindings, but there were also a few books with holes in the leather-lined and ornate binding covers through which an iron chain had once passed. According to the catalogue, the library also had at least one book with a partially preserved chain. I asked to be shown it to me. The book is stored in a special box; Several heavy black chain links lie in a separate compartment, not where the book itself lies: this way the leather of the binding does not rub against the iron. The library staff were just as curious to look at this artifact as I was. This only confirmed my belief that the story of the chained book, key to the story of the bookshelf, needs to be told once again. The point is not only that it is interesting in itself, but also that it is a concrete example of the evolution of an artifact that can be used to explain how technology penetrates and changes our culture.

It's understandable that most of us think more about books than shelves. But there were also those who paid tribute to the infrastructure. Thus, Henry Banner, who worked for many years as an editor at the humor magazine Pak, wrote:

Having become rich and successful,

I got myself a bookcase.

But I won’t keep books in it -

I won't spoil the beauty.

Of course, books can ruin another bookcase, but sometimes it is the bookcase that does not suit the books at all and almost discourages them from being removed from the shelves. When I moved into my current office at Duke University, it already had bookcases - quite nice ones, with adjustable height shelves. Because the shelves, made of heavy particleboard and finished in walnut, are deep enough without being too long, they are so strong that they do not sag even under very heavy books. But they are not very tall, so I adjusted the shelves so that at the right height I had the maximum number of shelves with books of different sizes. The result was that the books were grouped by height and there was little space left on the shelves above them. Sometimes it is difficult to grab a hold of a book and pull it out from a shelf that is filled to capacity. One guide to caring for books includes a question to help determine if they are stacked too tightly on a shelf: “Can you grasp a book with your index, middle, and thumb fingers and then gently remove it without moving adjacent books from either one?” not on the other side? So I can’t; I have to follow some good advice from Martha Steward's Living magazine: "To remove a book from a shelf, push the books to the right and left of it and pull gently."

Often, when there is enough space above the book, they do this: they put their finger on top and gently pull the spine, pushing out and turning the book until it becomes possible to grab it by the sides. Living magazine does not approve of this: “Never grab the spine with your finger.” If books are placed too tightly, you can break a nail or tear the binding, which is possibly even worse. One 19th-century “advice book” says: “Never remove a book from a shelf by the spine; do not dry them over the fire and do not sit on them, for “books are our good friends, their advice is useful to us, and they do not reveal our secrets.”

But inventor Charles Coley from Culver City, California, looked at books and bookshelves from a mechanic's point of view. He studied the problem of removing a book from a shelf and discovered that before him “no satisfactory solution to this problem existed.” In 1977, he received a patent for an “apparatus for retrieving books.” This is something like a wooden board on springs, located behind a row of books, across the back wall of a bookcase. It works on the action-reaction principle. To move a book out of a row of books, you need, contrary to the obvious, to press it into the back wall of the cabinet. This will compress the spring behind the board and the elastic force will push the book out. (This device works the same way as hidden latches on cabinet doors: you have to press down on the door to open it.) Like many inventions, the Kolya machine takes some getting used to; however, if the books are too dense on the shelf, it may simply not work.

In such circumstances, putting a book back on the shelf is no easier than returning a sardine to a tin. The bookshelf, apparently, does not tolerate emptiness, so the empty space that is created after removing a book from the shelf is rarely enough to easily return the book to its place. The book in this respect resembles an air mattress after use, or a map of the area that seems impossible to fold as intended. By opening a book and closing it, we seem to change its size. It no longer fits where it was. We have to wield it like a wedge, pushing aside the once obedient neighbors so that the book can finally establish itself in its rightful place. The book I'm trying to squeeze onto the shelf, of course, rubs against the neighboring books and pushes them deeper. If there is enough space above the books, it is quite easy to align them by hand. But in my office, it's not easy to reach between the books and the next shelf to line up all the spines. There is only one way out: you have to move the entire row deeper. But I also cannot bring all the books to the back edge of the shelf in this way: after all, they differ in width, which means that there will be no even row of spines. Over time, so many books end up pushed to the very back that the entire row has to be taken out and placed again closer to the front edge of the shelf.

It didn’t bother me that the books were deep on the shelves, because I liked there to be about five or six centimeters of free space from the front edge to the spine. I can't say exactly when I started arranging books this way or why. But at least I don't remember putting them exactly along the front edge, unless the widest book was the same width as the shelf itself. In this case, if I wanted an even row of spines, I had to push all the books forward. I started experimenting with leading edge alignment while working on this book. Before this, it seemed to me that a few centimeters of empty space in front of the books was quite natural and desirable; after all, even behind the books there are several centimeters of empty space. Thus, the books were centered on the centerline of the shelf, and the supports bore almost the same load. From a purely constructive point of view, it looked neat and correct. In public libraries, the aisles between rows of shelves are often narrow, and it is often impossible to see which books are on the edges of the shelves if they are pushed too deep, but in my home and office there is an empty wall opposite the shelves, and the distance to this wall is greater, than the width of a public library aisle. I can step back and look at them. If the books are at the very front edge of the shelf, then it seems as if the closet is too small for them (like a suit that has become too small), and its upper part outweighs the lower. Also, if the books are at the very front, the cabinets look two-dimensional: they have no depth, they resemble wallpaper. Where there is some free space above the books, there is certainly depth, but the top line is uneven, and the shadows falling on the books give their rows an even less neat appearance.

Since the books are pushed back on my shelves, there is a little free space in front of them where I can store all sorts of small trinkets like pencils and envelope cutters. All this seemed quite reasonable to me, until one day a writer came into my office. He was surprised at how my books were displayed, saying that he himself always put them at the very front and thought that this was the way to do it. I couldn’t give him a definite answer then, and I still can’t now. But I have since learned that the literary critic Alfred Kazin always moved his books to the back of his shelves so that he would have a place to put photographs of his grandchildren and to put the books he was currently reading. As with many issues of design and human interaction with technology, arguments can be made for both solutions. But in any case, I was glad that my guest was interested in how my books were arranged: it meant that I was not the only one thinking about bookcases and their use. But how and in what direction do such thoughts develop?

A book on a bookshelf is a thing that needs to be taken off the shelf and read. A bookshelf under a book is the kind of thing that you hang up and forget about. One object serves another or dominates another - this is the generally accepted logic, and a subordinate object rarely gives us cause for thought. But all people and objects - both ordinary workers and high-ranking persons - can tell some stories. And much more often than you might expect, these are exciting stories with unexpected twists and containing valuable information.

Is there anything whose form and purpose are more obvious than a bookshelf? It seems that the idea of ​​putting books on a wooden board is as old as books themselves. It seems that common sense and the laws of gravity dictate that the shelf be flat and horizontal. And the fact that books on a shelf should stand vertically, proudly straightening their spines, like a platoon of cadets, isn’t it self-evident for any library, large or small? We are confused by the portraits of Renaissance scientists: their offices are quite neat, but their books are anywhere but on the shelves. And if, nevertheless, they are located on the shelves, then in any way, just not vertically and not with the spine facing out. Isn't the vertical arrangement of a book on a horizontal shelf a law of nature? If not, why not? How and when did our present way of storing books become an almost universal practice?

The story of the bookshelf cannot be told without telling the story of the book, its evolution from scroll to manuscript and from manuscript to printed volume. One should not think that all these are dark deeds of the past, in no way connected with life in the new millennium. On the contrary, this information is incredibly important for understanding the history of civilization. It allows us to understand how technology is developing today and make predictions about the future (which will be much more similar to the present and past than we are usually led to believe).

Looking at a bookshelf (like any other object) with a fresh, unbiased look is useful in itself: in particular, this is how we understand the world and interact with it in a new way. Since books and the shelf underneath them are inextricably linked, if we focus on a hitherto forgotten bookshelf, we will be able to look at the book differently - turn it upside down, so to speak. When we look with a new eye at such a familiar thing as a book, we see a completely different object, the qualities of which distinguish it from all other things in the world and at the same time make it similar to many things that we know.

If there are only two books on the shelf, then they stand in an awkward position, like wrestlers in the ring. Three books on the shelf are reminiscent of basketball, when two defenders squeeze an attacking player. If there are even more books, then they look like schoolchildren playing leapfrog in the schoolyard. But more often than not, the shelf that is not completely filled with books is a commuter train, where passengers lean on each other and balance in precarious positions, although they are hampered by the acceleration of traffic.

A book on a bookshelf is an interesting thing. If it is not thick enough, it will not be able to stand on its own. A thin book, which is not supported by anything, continually falls in one direction or another - just like some weakling on the beach, who himself is embarrassed by his own frailty, and a thick book, which has no neighbors, swells: maybe , she’s bursting with pride or it’s the typographically-covered pulp that’s to blame, because the heavy pages bend the spine and push the covers apart, like a powerful sumo wrestler crouching in front of an opponent with spread legs: come on, push.

Ann Fadiman, author of the excellent collection of essays on books, Ex Libris, tells how she lost a 29-page pamphlet “so thin that its bright red spine could not accommodate the title.” This brochure “was lost between two plump neighbors, like a thin blouse in a crowded wardrobe that you can’t find for months.” In another essay, she explains why she prefers a bookcase to a wardrobe: “When my brother and I climbed through our parents’ bookcases, it gave us much more food for wild fantasies about their tastes and desires than studying their wardrobes. If you want any sense, look at the shelf.”

Books spend a lot of time on shelves. They seem to be waiting on the side of the road for someone to come up to them and offer them something to do. Books are ladies without gentlemen at a ball who stand against the wall and support each other; Only neighbors help each one maintain her position. The books are like the character in the movie Marty, who ended up in the same place every Saturday. Books in dust jackets are a queue at a bus stop, passengers buried in newspapers. Books are like bandits at an identification parade at a police station: they all match the signs, but the witness will only point to one. Books are what we are looking for.

Some books are private homes, filled with essays and articles on a single topic; some are anthology apartment buildings. The books on the shelf are the row houses of Baltimore, the clustered houses of Philadelphia, the townhouses of Chicago, the mansions of New York; in front of them is a narrow sidewalk, behind them are courtyards that are visible only to the owners. The stepped roofs form a common silhouette - a diagram of destinies, an urban landscape. As in all cities, passers-by walk along the sidewalk about their daily business and hardly see either individual buildings or their inhabitants. We may not notice a number of books at all until we start looking for some title, code, or specific address.

Not every book is doomed to get lost among others, to blend in with the crowd. Bestsellers are shining stars. But no matter how many famous or outstanding books are on a bookshelf, and no matter how many paparazzi are trampling around it, the shelf itself is a doormat. The shelves are the infrastructure of the library, a bridge on a country road and a local highway from point A to point B; and new expressways have already been built nearby, paving the way for the information highway (8) .

Bookcases are the main furniture in offices, bookstores, and libraries. A bookshelf is a floor on which books stand; the bed they sleep on until the reader-prince wakes them up or the talent scout promises them a stellar career. Books open the hearts of readers, and the shelves languish in frustration.

What are bookshelves waiting for? Of course, books. It rarely happens that someone fills an entire shelf in one fell swoop - unless, of course, the library belongs to a juggler who can throw a box of cigars in the air between two others, and then keep the whole structure in balance and the audience in admiration. This trick can be done with books, but not with an entire shelf! We usually put on our shelves a few books at a time or one or two that we received as a birthday gift or that we just bought. The bookshelf is not always full. This may be a joy for librarians, but it is a burden for bibliophiles: they prefer it when the shelf cannot be seen under the books.

A bookcase that is not completely filled with books is like the notebook of an absent-minded student: half of the lines in it remain free. If the closet is half full, then it is, of course, half empty. The books in it tilt left and right, forming the letters M, N, V ​​and W between clusters of vertical (and not so vertical) I's.

Although shelves are always ready to support books from below, they cannot always support an unstable book from the side. For tall books or for short books, bookends may (or may not) be suitable - curious devices that, in theory, should hold books back like a dam. But sometimes, as happens with dams, the bookends shift and collapse; gaps form in the once virtually monolithic façade of spines, and entire groups of books fall to the side in unsightly heaps. Before us, as in a video game, is the eternal conflict between the movement “up and down” and the movement “right and left,” between the obelisk and the sleigh (9) - both objects are subject to the force of gravity, but each in its own way. Gravity - the very force thanks to which bookends perform their function - determines the verticality of books. But the same force also acts in the horizontal plane. The frictional force caused by the weight of the book holder depends on it - a force opposite to that which causes sliding.

Contrary to popular belief, the simplest mechanism is not a wedge, but a block. A Victorian manual for furnishing a home library states that the "best device" that will make books "stand upright is made from a wooden cube six inches on a side, sawn in half diagonally." Bookends (many of which are simply carved blocks) create horizontal pressure that keeps books from falling. It's all about friction, of course, but as with any mechanism, the pressure that a bookend can withstand is limited, because the friction that occurs between the bookend and the shelf is also limited. The heavier and taller the holder, the better, and the rougher the contacting surfaces, the better. There are probably no other ways to improve the functionality of a book holder.

Some bookends have a thin metal base that sits under the first few books in a row: the weight of the books will provide pressure, which will then affect the friction between the holder and the shelf. Some holders are made from a sheet of steel, stamped and bent at the desired angle - a simple and clever solution. Such holders were patented in the 70s of the 19th century and have since become widespread, but they are not always suitable for a home library: they may not be strong enough to withstand the pressure of heavy books and maintain an upright position. This principle is implemented much more elegantly in such holders, where the vertical part is made of pleasant-looking wood, and the horizontal base is made of durable metal. My wife and I once found such bookends in a store in Indiana. The beautiful wooden strips were inlaid with almost invisible small ceramic mosaics, and the base was a heavy plate of galvanized metal; a thin foam rubber sole was glued to its underside to increase friction between the plate and the bookshelf. These bookends do their job well: they always stand upright and hold books in the same position. Alas, nothing is perfect: a base that is too thick gives the holder stability, but also raises the books on it three millimeters above the shelf level. It's hard not to notice. A gap forms under the books, which attracts attention. In addition, it rarely happens that the last book, under which the base of the holder lies, fits completely on it. Thus, it appears to stand on two steps, and the spine is noticeably deformed, since one cover of the binding is higher than the other. (The best holders are made from unnecessary books: only the bindings are left from them and these bindings are filled with something heavy. But many book lovers will not even want to hear about such barbarity. Book holders are also made from hard wood or stone: on one side they have cut out the “roots.” Such holders most often do not cause irritation.)

Among the most impressive bookends in my collection is a 635mm piece of real steel rail (funny how rails are often used as a metaphor for infinity). Of all my holders, this is the most massive; I glued a piece of felt to the bottom end to prevent the steel from scratching the shelf. Even the heaviest books are unable to move him from his place. But tall books sometimes topple it: the very shape of the rail means that its upper part is heavier than the lower. I have never come across an ideal book holder and it is unlikely that I will ever come across one. For every advantage there is a disadvantage, sometimes just as significant. This is the nature of man-made objects: to increase their advantages and reduce their disadvantages - this is the goal of engineering and design in general.

A board attached with brackets to the wall often acts as a shelf. In hardware stores, this is what is usually sold under the name “bookshelf.” When such shelves are placed one above the other, they are most often not closed at the ends - therefore, they need some kind of book holders. Sometimes the brackets of the shelf located above serve as limiters. For maximum effect, you can place a book on the bracket exactly at the height of the shelf. There is another option, in which the books themselves become the limiters: either some particularly thick volume holds back the neatly displayed books, or some of the books are placed horizontally on the shelf and provide the necessary mass, which a silent mechanism converts into friction when required. But everyone knows that if a long row of books begins to tilt, there is hardly a holder in the whole world that will provide sufficient friction: a stormy stream of books will rush from the shelf.

Bookstores of the late Victorian era sold such bookshelves. They were made from light boards and steel rods. These shelves needed to be hung on the wall.

If the shelves are not attached to the wall with brackets but are built into the bookcase, bookends may or may not be needed. If books fill the entire shelf, holders are no longer needed: vertical boards take on their role, and the books themselves become holders for their neighbors: one historical study props up another, a novel kisses a novel. Thus, a shelf in a bookcase is not just a horizontal board, but a board with vertical stops. Vertical boards, unlike book holders, allow you to easily squeeze some books between others: not only the friction force is involved here. If the shelves themselves are strong enough and can support the weight of the books, then the closet will always be able to accommodate everything that can be crammed into it.

Bookends should not slide on the shelf, but this does not apply to books. In my office, the wooden bookcase is painted with a cream colored semi-gloss paint. The shelves are dotted with marks from the bindings of the previous owner's books (mostly red and blue). I suspect he wanted to quickly get his books out of their boxes or off the floor and put them on the brand new shelves that he probably built and painted himself. He didn't wait for the shelves to dry properly. As a result, some paint from the bindings remained on the sticky surface of the shelves.

When a friend of mine arranged his library on freshly varnished shelves, he noticed that some books were more difficult to remove from the shelves than others. The worst thing was the massive volumes on engineering. He reasoned that sliding was hampered by friction arising between the shelf and the binding. He decided to wax the shelves like skis and polish them to a shine: after that it became easy to remove the books.

One professional bookcase designer solved the problem of friction between books and shelves differently by painting the shelves with automotive grade paint, which has “tremendous impact resistance and makes books slide easily.” And for some book designers, the physical characteristics of a book are more important than its ease of use: In 1853, inventor Charles Goodyear published a book printed on rubber pages and bound with rubber. This volume most likely stuck to any shelf and to neighboring books, like a tire on asphalt.

What is a book or bookshelf? As is often the case, the answer depends on the definition, and definitions change over time. Perhaps there is a bibliological analogue of the biological law, according to which ontogenesis in general terms repeats phylogeny; at least the similarity is sufficient for us to be able to utter that familiar ringing phrase here. Sometimes, especially if we are young, we make bookshelves for ourselves, which are not always strictly horizontal and vertical, but this is unintentional. As children, we build bookcases out of anything - for example, we turn a wooden orange box on its side, and we can put another box on top. Everyone knows that thin children's books never stand upright on their own; children put them on the shelves at random. But if you simply place a book on a horizontal surface, this surface will not become a shelf. If the books are on the desk - even even neatly, between the holders - the table does not turn into a shelf. Books on the windowsill are just books on the windowsill.

But it is the books that make the board a bookshelf, and the drawer a bookcase. Until books appear, boards and boxes remain boards and boxes. As we grow older, our tastes change. Many students have gone through the bricks and planks stage. Such shelves have an important advantage: they are easy to transport if the owner often moves from place to place. But at some point, most of us have the desire to have real shelves created specifically for storing books. We move up in our careers, we earn more and more, and now we want the best, built-in bookshelves in the house, preferably in a real office or even better - in the room that belongs to our books, that is, in the home library.

In the biography of Edward Bernays (a brilliant advertising man who took on everything from Dixie cups to Mack Trucks and was called the father of PR), it is written that built-in bookshelves became popular with architects, contractors and interior designers in 1930s, when Bernays was commissioned by publishers to boost book sales. According to one version of the story, he asked "respected and famous people to speak about the importance of books to civilization" and then persuaded those in charge of furnishing houses to install bookcases in them. The owner of the house had no choice but to start buying books: Bernays agreed with the aphorism that he is said to have carved on a wooden board: “Where there are bookshelves, there will be books.” But not everyone needed shelves that much. Ann Fadiman, whose parents owned about seven thousand books, writes: “As soon as we moved into a new house, a carpenter would come and make us shelves about a quarter of a mile long. When we left, the new tenants immediately removed these shelves.” When Thomas Jefferson's books were brought into the devastated Library of Congress after the Washington fire, the shelves, which were actually pine boxes that could be stacked on top of each other, had special covers nailed to the front to keep the books from falling out.

A small homemade bookcase: a wooden box with a shelf in the middle, bookends with slots nailed to the sides. Such a cabinet can be moved from place to place without even removing most of the books

During the Renaissance, works of art and various collections were displayed on all kinds of shelves. At the beginning of the 19th century, James Nesmith, a Scottish engineer and inventor of the steam hammer, wrote about his artist father, who moved from his workshop to another place: “On the walls and shelves of his study there are many objects of art and ingenious inventions, and almost all this is the work of his own hands.” This tradition is still alive today among collectors: quite often you can see in a house a room with shelves on which there are all sorts of things - from model trains to dolls; However, we won’t find a single book here. (In the house of an enthusiastic collector, there are probably various books with the addresses of art dealers and antique dealers, catalogs of advertisements for buying and selling, reference books with model numbers and prices, but all this, most likely, is in the bedroom: corner tables and even the corners themselves turn into a kind of office where business literature is stored, which the collector looks through before going to bed.)

In one gift album with magnificent photographs of the desks of famous people, mostly writers, you can see the office of Admiral William Crowe Jr. At the time of filming, he was Chairman of the US Joint Chiefs of Staff. Behind his desk is a luxurious bookcase that spans the entire wall, and on the shelves is a collection of hats, mostly military. These are hats, caps, helmets from all over the world, but there are no books in the closet. (If you look closely, you can see several books in the photo: it seems to be a desk dictionary and “Famous Quotations” (10), but they are as inconspicuous as the unblinking eyes of a palace guard, over which a ceremonial bearskin cap is pulled down. However, they will immediately attract attention the attention of a child, and the same thing happens with Admiral Crowe's books as soon as we see them.) On the shelves behind the desk of illustrator David Macaulay there are rows of toys, models, a variety of objects - everything in the world except books.

Most of us still put books on our bookshelves, and this is what we are talking about in our story, in which we will definitely have to touch on the history of the book - a subject that is deceptively simple, but in fact incredibly complex. Let’s immediately agree on the terms that denote different parts of the book here. The back cover is the part that comes into contact with the table when we place the book on it with the title facing up, so that it can be opened and read. When a book stands vertically on a shelf, the part in contact with the shelf is called the bottom edge, and the opposite part is called the top edge. The edge that is pushed inward is called the front edge - today it sounds paradoxical, but once upon a time it was he who looked outward. Finally, the part of the book that we see when looking at a shelf filled with books is called the spine. For many centuries, books were placed on shelves with the spine facing inward. In the history of the humble bookshelf, this is one of the most curious facts. Facts like these, and there are many of them, are what make this story interesting.

The history of the bookshelf and the way books are stored on it is the history of an object that acquires meaning only in context, only through use. Will a horizontal board be a bookshelf if there are no books on it? This question points to the defining difference between technology and art: technology must always be judged on the basis of utilitarian considerations, while art can be judged on the basis of aesthetics alone. The most beautiful bridge you can't drive on isn't a technological achievement, and it's hardly a work of art either. Even a very beautiful bookcase that collapses under the weight of books is not a bookcase, but an engineering failure. Can you say that a tree is making noise if no one hears it? Can we say that “empty shelf” is an oxymoron?

The evolution of the book and the evolution of the bookshelf are truly inseparable, and both are examples of the evolution of technology. Technological factors related to materials, functions, economics, and use influenced the appearance of books and book furniture more than literary factors. So, the evolution of the bookshelf is an example of technological development. But technology does not exist without the social and cultural context in which it operates and which, in turn, is significantly influenced. Therefore, the history of such a technological product as a book or bookshelf cannot be fully understood without understanding those aspects of it that, at first glance, are not related to technology.

If we describe how the methods of making a book, caring for it, and storing it have changed over the past two thousand years, an interesting and simple path to understanding the development of technology in principle will open before us. It will also help us better understand modern technologies, the development of which is so closely linked to our own development that we have difficulty noticing anything beyond the superficial changes that occur in everyday life. If we can better understand the mechanisms of technological evolution, we will be able to better understand what is happening to technology now, and thus predict what to expect from it in the future. Such insight is always valuable, whether we are investing in securities, creating and selling new products, or simply wanting to learn more about how the world works.

Annotation: Techniques for solving problems with restrictions on the sequence order or selection order are given. Particular solutions are given and general formulas are given. Problems involving the displacement of elements and pairs of elements are considered.

Problems with order restrictions

Until now, we have considered problems in which no restrictions or additional conditions were imposed on the order of elements in combinations. Or (as in combinations) the order was not taken into account at all. Let's consider problems with a constraint.

Problem 1. The tamer of wild animals wants to bring 5 lions and 4 tigers into the arena, but it is impossible for two tigers to follow each other. In how many ways can he arrange the animals?

Let's denote lions with the letter L. There are 6 places for tigers.

L 1 _____L 2 _____L 3 ____L 4 _____L 5 ______

Lviv can be located! In ways, that is, 120. On six places for tigers, they can be arranged in ways.

Total number of ways.

For the problem in general form, if there are: tigers and lions.

But since That

This is only possible provided that

Problem 2. A staircase is being built from point to point. Distance . The height of the step is 0.3 m, the width is 0.5 m or a multiple of 0.5 (Fig. 8.1). In how many ways can you build a staircase?

From the condition it is clear that the staircase must have, and there are 10 places where a step can be installed: and one extreme one.

Therefore, you need to choose 5 places for the step out of 10: in ways.

Construction options are shown in Fig. 8.2.

In general: if there are steps, then the staircase can be built in different ways.

This task is similar to the previous one; a tamer cannot place two tigers, and a builder cannot make steps of double height. But there is a significant difference: all the animals are different, but the steps are the same, so the builder has less choice.

A generalization of the ladder problem (encrypt the ladder with 1 and 0.....) can be the following: in how many ways can zeros and ones be arranged so that two ones do not stand next to each other?

There are ways to do this.

Restrictions on the order of selection

Problem 1. There are 12 books on the bookshelf. In how many ways can 5 of them be selected so that no two of them are adjacent?

Let's encrypt the choice of 0 and 1: we assign a 0 to each book left, and a 1 to each selected book. Thus, we have 5 ones and 7 zeros and the problem is reduced to the previous one.

In general: If there are books, but books are selected that are not nearby, then this can be done

Problem 2. There are 12 knights sitting at King Arthur's round table. Each of them is at enmity with his neighbor. You need to choose 5 knights (for example, on an expedition to free an enchanted princess), and so that there are no enemies among them. (Fig. 8.3) How many ways can this be done?

The difference from the previous task is that the knights sit not in a row, but in a circle. But it can easily be reduced to the case when knights sit in a row. To do this, let's take a knight, for example Sir Lancelot, and open the circle. All selected combinations fall into two classes: one involves Sir Lancelot, the other does not. Let's count how many combinations are included in each