December 26, 2015 at 04:14John Horton Conway: Life Is Like a Game
- Transfer
John Horton Conway claims to have not worked a day in his life. This excerpt from the biography “Genius at the Game” shows what serious mathematical theories, such as surreal numbers, may emerge from entertainment and games.
Gnawing the index finger of his left hand with his old broken-off British teeth, with swollen senile veins, with an eyebrow thoughtfully frowned under long-cut hair, the mathematician John Horton Conway spends his time thinking and theoretical research without regret. Although he will argue that he does nothing, is lazy, and plays with toys.
He works at Princeton, although he gained fame in Cambridge (first as a student and then professor from 1957 to 1987). Conway, 77 years old, claims to have not worked a day in his life. He means that he spends almost all his time on games. And at the same time, he is a Princeton professor of applied and computational mathematics (now honorary). Member of the Royal Community. And a recognized genius. “The title 'genius' is often misused,” says Percy Diaconis, a Stanford mathematician. - John Conway is a genius. Moreover, he can work in any field. And he has a flair for all sorts of unusual things. It cannot be put into any mathematical framework. "
The haughty atmosphere of Princeton, as it were, is not entirely suitable as a base for such a playful person. The buildings are built in the Gothic style and are covered with ivy. In this environment, a well-groomed aesthetic does not look old-fashioned. And in contrast, Conway is sloppy, always with a mysterious face on his face, a cross between Bilbo Baggins and Gandalf. Usually he can be found milling around in the mathematics department on the third floor in a common room. The department is located in the 13-story Fine Hall building, Princeton's tallest building, with rooftops from Sprint and AT&T operators. Inside, the number of professors is approximately equal to the number of students. Usually in the company of a student, Conway settles down either on one of the couches in the common room, or in an alcove by the window in the corridor in one of two armchairs looking at the board.
(paraphrase from the comedy "How do you like it": "Here is a poor virgin, a duke, a nondescript little thing, a duke, but actually belonging to me").
Conway’s contribution to mathematics includes countless games. Most of all he is known as the inventor of the game "Life" in the late 1960s. Martin Gardner, who led the column in Scientific American, called it “Conway’s most famous brainchild.” This is a game about a cellular automaton - a machine with groups of cells that evolves in steps in discrete time. Cells transform, change shape and evolve into something else. Life is played on a grid where cells resemble microorganisms examined under a microscope.
Rules of the game
Strictly speaking, this is not quite a game. Conway calls it "endless game without a player." The musician and composer Brian Eno recalls how he once saw a demonstration of this game on the monitor of the Exploratorium Museum in San Francisco and experienced an “intuitive shock”. “The whole system is so transparent that it should not bring surprises,” Ino says. “But there are enough surprises there - the complexity and organic nature of the evolution of point forms completely exceeds the predictions.” And, as an announcer in one of the episodes of the Steven Hawking television show Grand Design said: “It is quite possible to imagine that something like the game“ Life ”with a few simple laws can reproduce quite complex things, and maybe even intelligence. This may require a grid of millions of squares, but this will not be surprising. There are hundreds of billions of cells in our brains. ”
“Life” became the first cellular automaton, and remains the most famous. The game led to the fact that cellular automata and similar simulations began to be used in complex sciences, where they simulate the behavior of anything, from ants to traffic jams, from clouds to galaxies. And she became a classic for those who like to spend time aimlessly. The sight of groups of game cells changing on the computer screen turned out to be dangerous addiction for students of mathematics, physics and computer science, as well as for all people who had access to computers. In one of the military reports of the US Army, it was claimed that hours spent watching the Life game cost the government millions of dollars. Well, at least that's what the legend says. They also claim that when in the mid-70s the game began to quickly gain popularity,
Overview of life forms that can be found in the game / from a letter to Martin Gardner [clickable]
When vanity rolls on Conway (and this happens often), he opens some new book on mathematics, finds an alphabetical index, and is annoyed that his name is most often quoted only in connection with the Life game. And besides her, a huge number of his ideas, which expanded mathematics, are very diverse. For example, his first love is geometry, and as a result, symmetry. He stood out by discovering something called the "Conway constellation" - three sporadic groups in a family of such groups in an ocean of mathematical symmetry. The largest group is based on the Lich lattice, which is the densest packing of spheres in 24-dimensional space, where each sphere is in contact with 196560 other spheres. He also shed light on the largest sporadic group, the “Fisher-Gris monster,” in the Monstrous Moonshine hypothesis. And his biggest achievement, at least in his own opinion, the discovery of a new type of numbers called "surreal numbers." This is a continuum of numbers, including all real numbers (integers, fractional, irrational), and besides them all infinities, infinitesimal quantities, etc. Surreal numbers, by his definition, are "endless classes of strange numbers hitherto unknown to man." Perhaps they can describe everything from infinite cosmos to infinitesimal quantum quantities.
But the most amazing thing is how Conway came across them: playing and analyzing games. Like in Escher's mosaic, where the birds turn into fish. Focus on white and you will see birds. Focus on red and you will see fish. Conway watched a game like go, and saw something else in it - numbers. And when he saw these numbers, he was impressed for several weeks.
On his best days in Cambridge in the 1970s, Conway, always in sandals, at any time of the year, entering the common room of the department of mathematics, usually announced his arrival with a loud clap of his hand on one of the beams in the center of the room. And this action was ringing. The ringing marked the beginning of a new day of games. Especially interesting was one of the games, fatball (Phutball, AKA philosophical football).
This game was invented by Conway, but he himself does not play it very well.
In fact, the rules of a fatball allow a player who has made a very bad move to ask “can I cry?”, And if he is allowed, he takes the move back and replayes. But even in this case, Conway does not manage to play this game well, and in general he does not play particularly well in any games - at least he does not win very often. Nevertheless, he participated in an infinite number of games in the common room, raising them to a level worthy of serious scientific research. But at the same time, he sometimes allowed himself to suddenly jump up, clutching a pipe under the ceiling and swinging back and forth on it.
These performances did not make him the chief acrobat of the department. Then Frank Adams, a topologist who was fond of mountaineering, who liked to climb under the table without touching the floor, walked around him. Conway found him frightening, impossible by a serious mathematician. A professor of astronomy and geometry, Adams had a reputation as a person who is hard to please, whose lectures are hard to listen to - and a person who is strictly related to himself. Colleagues believed that his ambition was based on his periodic nervous breakdowns. Adams worked like a man possessed, which was a concern for Conway. He was sure that Adams did not approve of his passive relaxed manner. And that made Conway feel guilty and think that he should be fired - and he thought about his wife and an ever-growing group of his daughters, who needed to be supported.
He married Eileen Howie, a teacher of French and Italian, in 1961. “He was an unusual young man, which attracted me,” she says. - John and I soon after meeting went to a restaurant, and I stood waiting for him to open the door for me. And he told me - well, come on in! Most young people opened doors, pulled chairs, and all that. But it did not occur to him. He thought differently. Here is the door, you are standing in front of it, why not open it? There must be logic in that. ”
After marriage, they had four daughters, with periods of one, two and three years. Conway remembered their birth years as the 1960th plus the Fibonacci numbers: 1960 + 2, 3, 5, 8 = 1962, 1963, 1965, 1968.
And Conway was not in vain worried about the possibility of losing his job. By 1968, he had achieved little. All he did was play games in the common room, invent and reinvent the rules of games that he found boring.
Conway plays Life in 1974.
Conway liked games in which physical moves take place. He constantly played backgammon at small rates (money, chalk, just for fun). However, he never reached the heights in this game. He often took risks, took doubles when this should not be done, and raised rates up to 64 times from the original, just to see what would happen. And he constantly talked about mathematics. For example, “Conway's piano task”, which asks: what is the largest object you can move around a rectangular corner in a corridor of a fixed width?
He was not so interested in winning backgammon as he was interested in exploring the possibilities available in the game. He liked to lose on purpose, staying behind the worst players. His rivals often lost their vigilance and began to lose. And then he made his move. Usually this strategy did not work. But sometimes he was lucky (chance is an element of the game of backgammon, and therefore it does not lend itself to rigorous scientific analysis), and then he made a dizzying victory.
While Conway got hooked on backgammon, his other colleagues were afraid to play them for a long time, while others refused at all, fearing that they would get involved in the game and their research would be abandoned. Others have argued that Conway is a bad example for students. But that was his plan.
One student was Simon Norton, a child prodigy who attended Eton College and received a degree from the University of London while he was in high school. Arriving in Cambridge, and already an experienced backgammon player, Norton easily fit into the company. He was able to carry out calculations very quickly in his mind, and became Conway’s protege, solving for him tasks that he could not solve. He followed all the tasks that everyone and everyone solved, peeping, eavesdropping, interrupting anyone with shouts of “untruth!” When he noticed an error with him. He also had an impressive vocabulary, which Conway liked. He was known for his ability to solve anagrams. For example, once someone shouted the anagram “phoneboxes”. Before anyone could even raise their heads and understand what was going on, Norton proclaimed: "Xenophobes!"
Mostly Conway played stupid children's games - Dots and Boxes, Fox and Geese, and sometimes he played them with children, mostly with his four daughters. And, of course, with their minions - often in the games that they invented to please their leader. Colin Voot invented the COL game, and Simon Norton invented SNORT, and both games consisted of coloring areas. Norton also came up with the game Tribulations, and Mike Guy retorted by issuing Fibulations - both games similar to him, one based on triangular numbers and the other on Fibonacci numbers. Conway invented Sylver Coinage, in which two players took turns calling different positive integers, but they could not be called a number that was the sum of any of the numbers mentioned above. The first player to call the unit was losing.
Many games have been featured in Winning Ways for Your Mathematical Plays, a book written by Conway and two co-authors, Alvin Burlekamp, a mathematician at the University of California, and Richard Guy from the University of Calgary. The trinity of pioneers of game theory came together at the conference "Computers in Number Theory" in 1969. Conway - in the third row from above, the second on the right (with a beard). Alvin Berlekamp - fourth row from above, sixth to the right (also with a beard). Richard Guy - fourth row from above, ninth from left (with a striped tie). [clickable]
It took 15 years to write the book, partly because Conway and Guy loved to play nonsense and wasted Berlekamp's time. He called them a "couple of boobs." Nevertheless, the book became a bestseller, despite the fact that the need to print in color and in unusual fonts took so much money that there was almost no money left for advertising. The book was a tutorial on how to win games. The authors poured theories into it, as if from a cornucopia, and added many new games suitable for their theoretical research.
Conway wrote:
So they typed games without names, and names without games.
Sometimes Conway went on a visit to Martin Gardner, and they exchanged materials on mathematical entertainment. These were not necessarily games - they could be jigsaw puzzles and other entertainments for nerds. Take, for example, the Doomsday Algorithm, which allowed you to determine the day of the week for any date. And although he demonstrated this trick from a teenage age, an algorithm was invented during a visit to Gardner. Conway flew to New York and waited for his friend to pick him up from the airport. He waited, and waited ... But Gardner did not appear.
Gardner was more than two hours late, ran into the airport lounge, waving his arms frantically, with apologies and promises: “You will forgive me as soon as you find out what I found!” He was in the New York Public Library, where he found a note published in 1887 in the journal Nature: "How to find out the day of the week for any date." The article was written by Lewis Carroll. He wrote: “Having stumbled upon the following method of calculating in the mind of the day of the week for any selected date, I send it to you, hoping that it will interest any of your readers. I myself do not think very quickly, and it usually takes me 20 seconds to count this. But I think that it can take less than 15 seconds for a person who quickly counts. "
Gardner could not refuse to make a photocopy of the note, but there was a long queue for the copy machine. He stood in her, but she moved very slowly. By the time it became clear that he was late to pick up Conway, he had already stood for 30 minutes, and decided that another 15 minutes would be enough for him. He thought it was worth it, and knew that Conway would agree with him.
the end of
P.S .: calculation of the day of the week in the mind from Danov
Gnawing the index finger of his left hand with his old broken-off British teeth, with swollen senile veins, with an eyebrow thoughtfully frowned under long-cut hair, the mathematician John Horton Conway spends his time thinking and theoretical research without regret. Although he will argue that he does nothing, is lazy, and plays with toys.
He works at Princeton, although he gained fame in Cambridge (first as a student and then professor from 1957 to 1987). Conway, 77 years old, claims to have not worked a day in his life. He means that he spends almost all his time on games. And at the same time, he is a Princeton professor of applied and computational mathematics (now honorary). Member of the Royal Community. And a recognized genius. “The title 'genius' is often misused,” says Percy Diaconis, a Stanford mathematician. - John Conway is a genius. Moreover, he can work in any field. And he has a flair for all sorts of unusual things. It cannot be put into any mathematical framework. "
The haughty atmosphere of Princeton, as it were, is not entirely suitable as a base for such a playful person. The buildings are built in the Gothic style and are covered with ivy. In this environment, a well-groomed aesthetic does not look old-fashioned. And in contrast, Conway is sloppy, always with a mysterious face on his face, a cross between Bilbo Baggins and Gandalf. Usually he can be found milling around in the mathematics department on the third floor in a common room. The department is located in the 13-story Fine Hall building, Princeton's tallest building, with rooftops from Sprint and AT&T operators. Inside, the number of professors is approximately equal to the number of students. Usually in the company of a student, Conway settles down either on one of the couches in the common room, or in an alcove by the window in the corridor in one of two armchairs looking at the board.
Welcome! It's a poor place but mine own!
(paraphrase from the comedy "How do you like it": "Here is a poor virgin, a duke, a nondescript little thing, a duke, but actually belonging to me").
Conway’s contribution to mathematics includes countless games. Most of all he is known as the inventor of the game "Life" in the late 1960s. Martin Gardner, who led the column in Scientific American, called it “Conway’s most famous brainchild.” This is a game about a cellular automaton - a machine with groups of cells that evolves in steps in discrete time. Cells transform, change shape and evolve into something else. Life is played on a grid where cells resemble microorganisms examined under a microscope.
Rules of the game
Strictly speaking, this is not quite a game. Conway calls it "endless game without a player." The musician and composer Brian Eno recalls how he once saw a demonstration of this game on the monitor of the Exploratorium Museum in San Francisco and experienced an “intuitive shock”. “The whole system is so transparent that it should not bring surprises,” Ino says. “But there are enough surprises there - the complexity and organic nature of the evolution of point forms completely exceeds the predictions.” And, as an announcer in one of the episodes of the Steven Hawking television show Grand Design said: “It is quite possible to imagine that something like the game“ Life ”with a few simple laws can reproduce quite complex things, and maybe even intelligence. This may require a grid of millions of squares, but this will not be surprising. There are hundreds of billions of cells in our brains. ”
“Life” became the first cellular automaton, and remains the most famous. The game led to the fact that cellular automata and similar simulations began to be used in complex sciences, where they simulate the behavior of anything, from ants to traffic jams, from clouds to galaxies. And she became a classic for those who like to spend time aimlessly. The sight of groups of game cells changing on the computer screen turned out to be dangerous addiction for students of mathematics, physics and computer science, as well as for all people who had access to computers. In one of the military reports of the US Army, it was claimed that hours spent watching the Life game cost the government millions of dollars. Well, at least that's what the legend says. They also claim that when in the mid-70s the game began to quickly gain popularity,
Overview of life forms that can be found in the game / from a letter to Martin Gardner [clickable]
When vanity rolls on Conway (and this happens often), he opens some new book on mathematics, finds an alphabetical index, and is annoyed that his name is most often quoted only in connection with the Life game. And besides her, a huge number of his ideas, which expanded mathematics, are very diverse. For example, his first love is geometry, and as a result, symmetry. He stood out by discovering something called the "Conway constellation" - three sporadic groups in a family of such groups in an ocean of mathematical symmetry. The largest group is based on the Lich lattice, which is the densest packing of spheres in 24-dimensional space, where each sphere is in contact with 196560 other spheres. He also shed light on the largest sporadic group, the “Fisher-Gris monster,” in the Monstrous Moonshine hypothesis. And his biggest achievement, at least in his own opinion, the discovery of a new type of numbers called "surreal numbers." This is a continuum of numbers, including all real numbers (integers, fractional, irrational), and besides them all infinities, infinitesimal quantities, etc. Surreal numbers, by his definition, are "endless classes of strange numbers hitherto unknown to man." Perhaps they can describe everything from infinite cosmos to infinitesimal quantum quantities.
But the most amazing thing is how Conway came across them: playing and analyzing games. Like in Escher's mosaic, where the birds turn into fish. Focus on white and you will see birds. Focus on red and you will see fish. Conway watched a game like go, and saw something else in it - numbers. And when he saw these numbers, he was impressed for several weeks.
On his best days in Cambridge in the 1970s, Conway, always in sandals, at any time of the year, entering the common room of the department of mathematics, usually announced his arrival with a loud clap of his hand on one of the beams in the center of the room. And this action was ringing. The ringing marked the beginning of a new day of games. Especially interesting was one of the games, fatball (Phutball, AKA philosophical football).
Fatball rules A
game starts with one chip (“ball”), which is placed on the central crosshairs of a square grid - for example, for playing go. Two players sit on opposite sides of the board and take turns. At each turn, the player can either put one white chip (“person”) on any free intersection, or perform a sequence of jumps.
To jump, the ball must be near one or more people. It moves in a straight line (vertical, horizontal or diagonal) to the first free intersection behind a person. The person who jumped is removed from the field. If a jump is made, the same player can continue to jump, all the while the ball is next to at least one person - or it can stop moving at any time. Jumping is optional - you can instead place a person on the field.
The game ends when the sequence of jumps ends at the edge or behind the edge of the board closest to the opponent (“goal”), in which case the one who jumped wins. A series of jumps can take place exactly along the goal line of the opponent, without going beyond it. One of the interesting properties of a fatball is that every move can be played by any player.
This game was invented by Conway, but he himself does not play it very well.
In fact, the rules of a fatball allow a player who has made a very bad move to ask “can I cry?”, And if he is allowed, he takes the move back and replayes. But even in this case, Conway does not manage to play this game well, and in general he does not play particularly well in any games - at least he does not win very often. Nevertheless, he participated in an infinite number of games in the common room, raising them to a level worthy of serious scientific research. But at the same time, he sometimes allowed himself to suddenly jump up, clutching a pipe under the ceiling and swinging back and forth on it.
These performances did not make him the chief acrobat of the department. Then Frank Adams, a topologist who was fond of mountaineering, who liked to climb under the table without touching the floor, walked around him. Conway found him frightening, impossible by a serious mathematician. A professor of astronomy and geometry, Adams had a reputation as a person who is hard to please, whose lectures are hard to listen to - and a person who is strictly related to himself. Colleagues believed that his ambition was based on his periodic nervous breakdowns. Adams worked like a man possessed, which was a concern for Conway. He was sure that Adams did not approve of his passive relaxed manner. And that made Conway feel guilty and think that he should be fired - and he thought about his wife and an ever-growing group of his daughters, who needed to be supported.
He married Eileen Howie, a teacher of French and Italian, in 1961. “He was an unusual young man, which attracted me,” she says. - John and I soon after meeting went to a restaurant, and I stood waiting for him to open the door for me. And he told me - well, come on in! Most young people opened doors, pulled chairs, and all that. But it did not occur to him. He thought differently. Here is the door, you are standing in front of it, why not open it? There must be logic in that. ”
After marriage, they had four daughters, with periods of one, two and three years. Conway remembered their birth years as the 1960th plus the Fibonacci numbers: 1960 + 2, 3, 5, 8 = 1962, 1963, 1965, 1968.
And Conway was not in vain worried about the possibility of losing his job. By 1968, he had achieved little. All he did was play games in the common room, invent and reinvent the rules of games that he found boring.
Conway plays Life in 1974.
Conway liked games in which physical moves take place. He constantly played backgammon at small rates (money, chalk, just for fun). However, he never reached the heights in this game. He often took risks, took doubles when this should not be done, and raised rates up to 64 times from the original, just to see what would happen. And he constantly talked about mathematics. For example, “Conway's piano task”, which asks: what is the largest object you can move around a rectangular corner in a corridor of a fixed width?
He was not so interested in winning backgammon as he was interested in exploring the possibilities available in the game. He liked to lose on purpose, staying behind the worst players. His rivals often lost their vigilance and began to lose. And then he made his move. Usually this strategy did not work. But sometimes he was lucky (chance is an element of the game of backgammon, and therefore it does not lend itself to rigorous scientific analysis), and then he made a dizzying victory.
While Conway got hooked on backgammon, his other colleagues were afraid to play them for a long time, while others refused at all, fearing that they would get involved in the game and their research would be abandoned. Others have argued that Conway is a bad example for students. But that was his plan.
One student was Simon Norton, a child prodigy who attended Eton College and received a degree from the University of London while he was in high school. Arriving in Cambridge, and already an experienced backgammon player, Norton easily fit into the company. He was able to carry out calculations very quickly in his mind, and became Conway’s protege, solving for him tasks that he could not solve. He followed all the tasks that everyone and everyone solved, peeping, eavesdropping, interrupting anyone with shouts of “untruth!” When he noticed an error with him. He also had an impressive vocabulary, which Conway liked. He was known for his ability to solve anagrams. For example, once someone shouted the anagram “phoneboxes”. Before anyone could even raise their heads and understand what was going on, Norton proclaimed: "Xenophobes!"
Mostly Conway played stupid children's games - Dots and Boxes, Fox and Geese, and sometimes he played them with children, mostly with his four daughters. And, of course, with their minions - often in the games that they invented to please their leader. Colin Voot invented the COL game, and Simon Norton invented SNORT, and both games consisted of coloring areas. Norton also came up with the game Tribulations, and Mike Guy retorted by issuing Fibulations - both games similar to him, one based on triangular numbers and the other on Fibonacci numbers. Conway invented Sylver Coinage, in which two players took turns calling different positive integers, but they could not be called a number that was the sum of any of the numbers mentioned above. The first player to call the unit was losing.
Many games have been featured in Winning Ways for Your Mathematical Plays, a book written by Conway and two co-authors, Alvin Burlekamp, a mathematician at the University of California, and Richard Guy from the University of Calgary. The trinity of pioneers of game theory came together at the conference "Computers in Number Theory" in 1969. Conway - in the third row from above, the second on the right (with a beard). Alvin Berlekamp - fourth row from above, sixth to the right (also with a beard). Richard Guy - fourth row from above, ninth from left (with a striped tie). [clickable]
It took 15 years to write the book, partly because Conway and Guy loved to play nonsense and wasted Berlekamp's time. He called them a "couple of boobs." Nevertheless, the book became a bestseller, despite the fact that the need to print in color and in unusual fonts took so much money that there was almost no money left for advertising. The book was a tutorial on how to win games. The authors poured theories into it, as if from a cornucopia, and added many new games suitable for their theoretical research.
Conway wrote:
We invented a new game in the morning in order to serve as an application of our theory. And after half an hour of research, she turned out to be nonsense. Therefore, we invented another. Roughly speaking, ten times a half an hour in the working day, and so we invented 10 games a day. We examined them, sifted, and about one out of ten turned out to be good enough to be included in the book.
So they typed games without names, and names without games.
We had a "marriage problem." We invented the game, and if it was good, then there was a problem somehow to name it. If the name was not invented, we placed it in the "games without a name" folder. And then, scrupulous and precision-loving, Richard started another folder, "titles without games." All attempts to invent a name for the game ended with the invention of a bunch of different names that didn’t fit the right game, but were not bad in themselves. Therefore, they went to the folder "names without games." Each of the folders grew, and we rarely managed to “get married” records from two folders.
I remember the best name without a game. It sounds like “don’t call us, we’ll call you” (Don’t Ring Us, We'll Ring You; ring - “ring”, as well as “ring”, or “circle”). We did not get our hands on the invention of the game, but its essence is completely clear: the players would draw something on paper, and the goal would be to circle their opponent. For such a game, that would be a great name. But we never came up with the game itself.
Sometimes Conway went on a visit to Martin Gardner, and they exchanged materials on mathematical entertainment. These were not necessarily games - they could be jigsaw puzzles and other entertainments for nerds. Take, for example, the Doomsday Algorithm, which allowed you to determine the day of the week for any date. And although he demonstrated this trick from a teenage age, an algorithm was invented during a visit to Gardner. Conway flew to New York and waited for his friend to pick him up from the airport. He waited, and waited ... But Gardner did not appear.
At first I thought - okay, he will appear in five minutes. But I waited there for a very long time - I don’t know, maybe an entire hour. And I began to think: “What if he does not appear?”. I didn’t even have his phone. And even if I had, I did not know how to make a phone call in America. Therefore, I could only sit there and hope.
Gardner was more than two hours late, ran into the airport lounge, waving his arms frantically, with apologies and promises: “You will forgive me as soon as you find out what I found!” He was in the New York Public Library, where he found a note published in 1887 in the journal Nature: "How to find out the day of the week for any date." The article was written by Lewis Carroll. He wrote: “Having stumbled upon the following method of calculating in the mind of the day of the week for any selected date, I send it to you, hoping that it will interest any of your readers. I myself do not think very quickly, and it usually takes me 20 seconds to count this. But I think that it can take less than 15 seconds for a person who quickly counts. "
Gardner could not refuse to make a photocopy of the note, but there was a long queue for the copy machine. He stood in her, but she moved very slowly. By the time it became clear that he was late to pick up Conway, he had already stood for 30 minutes, and decided that another 15 minutes would be enough for him. He thought it was worth it, and knew that Conway would agree with him.
the end of
P.S .: calculation of the day of the week in the mind from Danov