MAT 007 I News WWW Edition
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MAT 007 I News, Wreckreational Math

World Wide Web Edition, Published Bioccasionally

[007 News]

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man to the bonds of Hell -- St. Augustine

Table of Contents:

From the Editors

This special edition of MAT 007 I News is a selected compilation of articles that have been written over the past eight years.

I'd tell you what the mission of our newsletter is, but instead, I have appropriately placed one of the Editor's Notes of Steve Sculac, the founder of 007 News, below. It's quite self-explanatory.

The idea of having 007 News online was an idea proposed by Andrew Irwin in 1992, during my first year as editor. Now with the World Wide Web, it has now been made possible and we couldn't be more pleased with the results.

The 007 News Home Page contains information on how you can subscribe to our newsletter for absolutely free. We also welcome any Letters to the Editor. In the meantime, I hope you enjoy what's in store for you below.

Joel Chan

Yes, once again, the MAT 007I group has put together another newsletter. For those who are seeing this publication for the first time, I'll begin with a brief explanation of what it's all about. Put succinctly, this newsletter attempts to combine two things -- mathematics and humour. The articles range from the very serious and informative to the witty and (hopefully) hilarious. The ultimate goal of the newsletter is to bring you some enjoyment in mathematics that you might not be able to get anywhere else.

At this point I'd like to encourage everyone who reads this newsletter and gets an idea, no matter how crazy it might be, to send it to us and, if it's at all possible, we will use it in a future issue. I hope that this sharing of ideas might inspire someone to something of value.

At this point I'd like to acknowledge Marie Bachtis who is responsible for the existence of this publication. We had a much smaller idea in mind when we first approached her, and she blew it up totally out of proportion to where now thousands of mathematicians are now exposed to our (shall I call it) work. Thanks Marie!

That's all I have to say for now. I hope you like what you read.

Steve Sculac

This was the editor's note from Volume 3 Number 1 of 007 News.

Geometry Corner with Martin Gardner

No doubt most of you will have heard of Martin Gardner. Mr. Gardner wrote the Mathematical Games column in Scientific American for twenty-five years. He is also the author of many books on science, philosophy, literary criticism, and, of course, mathematics. Our staff would like to thank him for his article, exclusively written for 007 News.

The following six-part problem was devised and answered by Karl Scherer, a computer scientist now living in Auckland, New Zealand. It has not previously been published.

  1. Take a square and cut it into three congruent parts.
  2. Take a square and cut it into three similar parts, just two of which are congruent.
  3. Take a square and cut it into three similar parts, no two of which are congruent.
  4. Take an equilateral triangle and cut it into three congruent parts.
  5. Take an equilateral triangle and cut it into three similar parts, just two of which are congruent.
  6. Take an equilateral triangle and cut it into three similar parts, no two of which are congruent.
Solutions are given in Volume 7 Number 2 of MAT 007 I News and the September 1994 edition of Math Horizons, the undergraduate magazine published by the Mathematical Association of America. How many can you get before you're forced to look up the solutions?

You Call This Employment Equity?

From the November 19, 1995 edition of the Toronto Star (Context Section)

(The Politics of Job Equity, Part 2 of 5, by Sandra Martin)

... The other basic question was just as troubling: Should employers base their employment equity on the number of designated group members they were hiring -- intake -- or on the representation of designated group members in their work force?

The same numbers can be counted in different ways depending on the results you want to achieve. For example, in 1993 at the University of British Columbia, women represented 20.56 per cent of all faculty. That sounds pretty grim, considering UBC has had an employment equity policy since 1990. But that 20 per cent figure is weighted down by the aging male professiorate, most of whom will reach retirement age before the end of the decade. In other words, they will be gone by 2000.

When you look more closely at the numbers, you can see that women accounted for 17.15 per cent of tenured appointments at UBC in 1993 and 32.99 per cent of tenure-track jobs. When you add those last two figures together you get 50.14 per cent. By doing nothing, women will eventually outweigh men as a percentage of total faculty at UBC.

This story is being repeated at universities across the country. [...]

(Editor's note: Watch for a response by the 007 Editor in the near future.)

A 007 exclusive: The secret memoirs of Bernard Mandeltrob

I'm keeping this hidden so that the truth may be known after I have long faded into obscurity. I regret that I may have delayed the progress of science for a short time, but that is the price the world must pay so that I can live the life of a scientific celebrity.

When I first began my study of "fractals", I was studying the properties of the equation z=z^2. The purpose of this was to see what would become of the value when subjected to round-off. Of course it is obvious that any value of absolute value less than 1 would eventually converge to 0 - expedited by the truncation errors that finite, physical means will impose.

It occurred to me, one day, that perhaps this property would be more interesting when performed using complex numbers. The interactions of the multiplications would be far more interesting. I was also interested in how adding a constant would perturb the system of errors. So, my now famous equation of z = z^2 + c -- where z and c are elements of the complex plane -- came to be. I decided to keep z initialized to 0 since, having taken high-school science, thought that only changing one variable would make things less confusing.

Such was not to be the case. When I tried a few cases by hand, the results seemed to be nothing more than random! Some values of c caused the value of z to explode to infinity, others caused the value of z to converge to a finite number of points, and others still caused the value of z to circulate without end, but without causing the rapid growth of z! In order to separate the bounded sequences from the unbounded sequences, I decided that producing a graph would be the easiest way of mapping the set. When I tried producing a graph by hand, it seemed that I was getting random points of convergence and divergence. I then tried producing a graph on a computer, but found the image even more bizarre. IBM somehow took interest in my work without me having to explain what it was. All I told them was that I was working on "fractals". You may be asking yourself, "where did this guy get a name like 'fractal'"? You may have heard that it means "fractional dimension" describing the possibility of the generated set being of infinite perimeter but finite area, etc. In fact, I coined the name fractal because it's the word "fractional" spelled erroneously! Get it!?! I was studying "the errors of fractional values"!

Now, my fame arose, quite accidently, after a nosy physicist at IBM saw my bizarre set on the screen. He exclaimed, "That looks like a feedback system I've been trying to model!". The physicists took to my "fractals" instantly! They saw it useful for simulating all kinds of phenomena they had measured in the lab. Well, the truth can now be told... The only reason these equations produce anything that looks like what physicists see in the lab is that physicists are constantly being victimized by round-off error! The physics community is using computer generated round-off error to simulate their own round-off errors! This is probably why physics has made so few advances after embracing this concept.

Now that there are no more accolades, I will have to say that more physicists should delve into pure mathematics, where the beauty of truth is not marred by round-off errors. I am sorry for allowing science to run astray for so long, but, I have enjoyed a comfortable life, I was admired by many, I sold a lot of books, and that's the way human nature is...

Silly Math Humour

A mathematics student had just finished his Ph.D. in Princeton, and he was looking for jobs. After a year with no success, he finally landed a job with the zoo as a zookeeper. One day, the bear in the zoo died. The zoo was facing the same financial crisis as the universities, and so they could not afford to buy another bear. So they asked the student to dress up in a bear costume and pretend that he was a bear. Well, the salary they offered was definitely an increase, and so he took this job. He was put into a cage, and with time he became very good at imitating a bear. But he had one worry. The bars between his cage and the next cage were loose. And in the next cage was a very ferocious looking lion. One day, his worst fears were realized, and the bar broke loose. The lion jumped through the bars, and ran up to the student. Extending his paw, the lion exclaimed, "Hi, I'm Phil, a physics major from Stanford."

What do you call a young eigensheep?

  • A lamb, duh!

    Games on Graphs

    By Phil Morenz

    The game Pong Hau K'i in Volume 2 Number 2 of MAT 007 I is a simple example of a much larger class of games played on finite graphs. Roughly speaking a graph is a set of points (called vertices) joined by lines (called edges). For what follows we will not need to give formal definitions, but for the interested reader two good introductions to graph theory are [1], [2]. We will only be concerned with finite simple graphs (finite number of vertices and edges; no edge joins a point to itself, any two points have at most one edge connecting them). Some examples of graphs: [5 Examples of Graphs Pictured]

    Notice that all of these graphs are finite but #3 is not simple. Notice that graph #5 is in two "pieces". We will only be concerned with connected graphs those with only one "piece". Now for the fun and games. The simplest game is called cop and robber. Given a finite simple connected graph, the cop starts at one vertex, the robber at another. At his turn each must move along an edge to an adjacent vertex. They alternate turns. The cop tries to land on the robber (or have the robber land on him!), the robber tries to get away. The robber moves first. Now for the questions! What is the smallest graph (i.e.: smallest number of vertices) so that a single robber can always evade a single cop, no matter where they start? (Hint: 3 is too small, 4 depends on starting positions, 5 is just right.) That was easy! A tree is a special kind of graph (#4 is an example) If you don't know the definition, check [1] or [2] or ask a computer scientist. Can you prove that if the graph is a tree the cop always catches the robber? Also not hard. Now suppose that we have two cops instead of one. Both move simultaneously. Can you construct a graph so that one robber can always evade two cops? One more question, one that I don't know the answer to, so I'll offer a prize of $1 (one loony!) for the best solution by an undergraduate, time limit, 1 month. The question is, what is the smallest (ie.: fewest vertices) graph such that one robber can always evade two cops (independent of starting positions)?


    1. Bondy and Murty, Graph Theory with Applications, North Holland, 1976.
    2. Harary, Graph Theory, Addison-Wesley, 1969.

    Paradoxical Notes

    [Editor's note: In this new feature we will discuss 'paradoxes' of a mathematical nature. We are always on the lookout for strange results; so send that weird idea in!]

    This week's topic is the Banach-Tarski paradox. We expect many readers to be familiar with it so we shall only provide a brief explanation. For the uninitiated, please note it will be mind-expanding, but probably not fatal. [Ed: The 007 and its affiliates will assume no liability for damaged brains, degrees or careers if you read any further.]

    The Banach-Tarski paradox states that if you take any object (say, the unit sphere) then it is possible to cut it into finitely many parts (five in fact), and put these pieces back together to get two copies identical to the original. There is no trickery here. The result is real and has been proven. It is left as an exercise for the interested reader to determine why no one has applied this theorem to their economic advantage. A simple corollary is that any nice set can be cut into finitely many pieces and reassembled into a desired number (either more or fewer) of identical copies.

    The editorial board of the 007 has found two applications of this theorem. Unfortunately it is used to explain, and not to cause events.

    We like to play tennis. At least, that's what we call it. We arrive at the appointed hour, and the chaotic dynamics begin. We have discovered that it helps to have lots of tennis balls. The only useful conjecture we have come up with during these sessions is:

    Conjecture: If tennis is played with a sufficiently large number of tennis balls, and if you don't keep score, then it is impossible to have the same number of balls at the end.

    It should be noted that even loud fluorescent orange balls can be Banach-Tarskied away. Generally, though, only the usual yellow-green ones Banach-Tarski into existence.

    Proof: Meet us Thursday mornings on the court. It will be clear how to apply the paradox.

    The second observation involves the formation of copy for the next issue. Invariably articles or other submissions that are supposed to appear, don't, and others (such as this one) appear in their place. The same can be said about the typographical errors, and even whole paragraphs that are just plain wrong. Mathematics has provided this insight into one of the "weird" forces of nature. The paradox is one of those driving forces which keeps the world interesting just when everything seems to be settling down into a nice understandable pattern.

    As we noted above, the "paradox" has been proven; so it isn't really a paradox at all. The trouble hinges on the fact that the proof merely demonstrates that the partitions exist (isn't the axiom of choice frustrating?), but gives no recipe for cutting. We propose that the nature of the cuts is such that they are only valid if not observable (in a Quantum Mechanical sense). The interesting thing about the examples above is that the paradox exerts its influence just when the events in the world get confusing. It seems clear that if there were enough people following the tennis balls (or fewer balls), that they would not Banach-Tarski. After all, highway 401 is almost as confusing as one of our games, but cars don't Banach-Tarski because there is a driver keeping an eye on each one.

    Is It Open?

    At a local high school, population 1000, all the students are lined up outside the locker room. Initially, all the lockers, numbered sequentially from 1 to 1000, are closed. The first student enters the room and toggles the state of every locker. (If it's closed then open it, but if it's open, then close it.) The second student enters and toggles those lockers starting at the second locker, counting by twos. Student three starts at locker three and toggles every third one. Each student does this in turn. Find a general expression specifying exactly which lockers are open at the end of the process.

    Setting Things Straight

    By Cathy Nangini

    Sometimes textbook mathematics does little more than confound you with abstract notions and vague ideas. So I have compiled a list of some basic concepts sure to set anyone straight!

    Groups. It is an inherent feature of groups that all its members have the ability to multiply things together. Conservative groups use the standard notion of mathematical multiplication whereby 2 * 3, say, is always 6. Then there are the Revolutionaries, who throw out all convention and embrace more radical platforms by defining multiplication to be anything they so desire. The integers, for example, could become a Revolutionary Group if they choose to define multiplication as addition instead. This could create a lot of group tension, and perhaps even a group war, as the Radicals pull away from their real roots. Incidentally, this explains why we never speak of a group of politicians, we say party. Politicians can neither add nor multiply as individuals, let alone as a group.

    Sets. The notion of the mathematical set is slightly confusing to the uninitiated topologist. Set theory, however, can be made relatively simple if we consider the following analogy. A set is much like your bank account. A non-empty set, like an account, must contain something. Bank accounts can be either open or closed, as it is with sets. But here we must be careful, because sets, unlike your bank account, can be both open and closed at the same time. This is just a mathematical paradox we shouldn't have to worry too much about. We must also remember that a set that contains zero contains something, but a bank account that contains zero means that you are broke. The other thing is that sets do not charge interest; that is why topologists do not work in banks.

    Subspaces. An important concept associated with vector spaces is that of rank. The rank of any space is determined by its dimension; that is, the number of linearly independent elements in the basis set. For subspaces, and more importantly, subspace messages, the rank of the transmitting party determines the priority of the message. So a Starfleet Officer of rank n, for example, will receive a subspace transmission of n dimensions, which means it is a very important message indeed. Those of infinite rank, however, are too great to be found on any starship. Hence these people are usually reserved for the Complex Plane...

    A Useful Guide to Reading Math

    Three Guys and a Large Number

    By Joel Chan

    This article appears in the February 1995 issue of Math Horizons, the student magazine of the Mathematical Association of America.

    In the late evening hours of April 26, 1994, it was announced that one of the most famous problems in cryptography, RSA-129, had been solved. A group of six hundred volunteers on the Internet led by Derek Atkins of MIT, Michael Graff of Ohio State University, Arjen Lenstra of MIT, and Paul Leyland of Oxford University carried out a job which required factoring a large number into two primes in order to crack a secret message and took eight months and over 5000 MIPS years, or approximately 150,000,000,000,000,000 calculations!

    RSA-129 is actually a 129-digit number that is used to decrypt (or decode) the secret message by the RSA algorithm. RSA is a public-key cryptosystem and is named after the inventors of the algorithm: Ron Rivest, Adi Shamir, and Leonard Adleman.

    What is a public-key cryptosystem? In the past, messages were encoded by a scheme called a secret-key cryptosystem. The sender and receiver of a message would have the same key (or password) in order to encrypt and decrypt messages. A problem of this system is that the sender and receiver must agree on this secret key without letting others find out. So in 1976, a new system called public-key cryptography was invented that took care of this problem. In this system, each person receives a pair of keys, called the public key and the private key. Each person's public key is published but the private key must be kept secret. This way, a user can encode a message using the intended recipient's public key, but the encoded message can only be decrypted by the recipient's private key.

    The RSA algorithm is a simple, yet powerful, cryptosystem. It's simple in the sense that the algorithm can be easily understood by any mathematician. In fact, here's how it works:

    First we translate the message into numeric form. For instance, suppose we want to send this message to your math professor: PLEASE DONT FLUNK ME. We can use a simple encoding scheme such as letting 01=A, 02=B, ..., 26=Z, and 00 be a space between words. So the numeric message becomes

    1612050119050004151420000612211411001305 = t.

    Actually, any text to numeral converter will do, since this is really not part of the RSA algorithm. But this means we have a secret-key cryptosystem in addition to our public-key encryption!

    So how are RSA public and private keys generated?

    Suppose Eric wants to send the secret message t to Darcy, his professor. Suppose you wanted to crack Darcy's message from Eric but you didn't know Darcy's private key. The obvious way to try to crack the code is by trying to calculate the two prime factors of the public modulus n. But that is also the beauty of the RSA algorithm. Currently there are no quick ways of factoring large numbers. This also means that RSA depends on the fact that factoring is difficult. Even though RSA-129 has been solved, a larger number takes exponentially longer to factor. So unless mathematicians are able to find relatively easy ways of factoring large numbers (which is unlikely in the near future), the security of RSA is safe.

    In fact, shortly after RSA was developed in 1977, Rivest, Shamir, and Adleman proposed a challenge to the scientific world to crack an encoded message given the public keys n = RSA-129 and e. In fact, they asserted that even with the best factoring methods and the fastest computers available at the time, it would take over 40 quadrillion years to solve. But with the exponentially increasing power of computers and advances in mathematical techniques in factoring large numbers -- a factoring scheme called the quadratic sieve was invented in 1981 by Dr. Carl Pomerance of the University of Georgia, the method used in factoring RSA-129 -- it would be somewhat surprising that it would only be 17 years later when the message would be cracked.

    Well, enough of the theory, and let's get to the gooey stuff: the calculations by the 600 miracle workers.

    The encoded message published was c = 9686961375462206147714092225435588290575999112457431987469512093081629822514570

    with the public key pair e = 9007 and n = RSA-129 = 1143816257578888676692357799761466120102182967212423625625618429357069352457338

    = 3490529510847650949147849619903898133417764638493387843990820577 * 32769132993266709549961988190834461413177642967992942539798288533.

    With these two primes, the private key is calculated to be d = 1066986143685780244428687713289201547807099066339378628012262244966310631259117

    Upon applying the decryption step, the decoded message becomes t = 200805001301070903002315180419000118050019172105011309190800151919090618010705.

    Using the 01=A, 02=B trick, the decoded message reads


    The definitions of the last two words are left to the reader as an exercise.

    For more information about RSA and cryptography, check out the World Wide Web Virtual Library.

    Dear Gwendolyn

    Dear Gwendolyn, I am desperate. I mop floors on weekends at an unnamed university in Ontario. Unfortunately, due to the dwindling funds available for sanitation, the department hasn't been able to maintain the upkeep of our equipment. I find that the width of my mop is shrinking by half each day due to the wear and tear I put on it. I asked an engineer friend of mine where it will all end. He, pulling out his calculator, said I'd end up with a point after just thirty days. My problem is that I still have mop up the same area! What happens in a month? What will I do then? Your devoted reader, Marty the Mopper.

    Dear Marty, your problems are manifold. First, you have an engineer for a friend. Please correct the situation. Second, your mop will never shrink to a point in finite time (even nicely). Third, even if it does become a point, then just follow a space-filling curve when you mop, and charge them for overtime.

    Dear Gwendolyn, I really don't what to make of this, but ever since I started taking this algebra course I have been having difficulties doing my math homework. It seems that everytime I sit down to do my work I feel like my head is going to explode. What is going on? Am I crazy? Sincerely yours, Vector Spaced

    Dear Spaced, don't be alarmed. Apparently you have caught a mild case of Dysfunctionism, caused by cerebral-infecting organisms of genus linearum algebratus. It is a condition known only to affect undergraduates, and it generally targets math students (all of whom, incidentally, have had to take courses in linear algebra). Those who have it often complain of sudden dizziness and confusion when approaching a mathematics text of any kind, accompanied by an overwhelming sense of doom. The best solution is psychological. By deluding the brain into thinking of anything but math until the last possible moment seems to be a good strategy. For instance, try covering your textbooks. That way, you will approach your desk unaware of the algebra lying in wait and, if you choose nice pictures, you will at least have something interesting to look at.

    Prof. Man-Duen Choi's Problem

    A math class of 20 students got their test papers back. The average was 54.0% (it was an easy test!). The professor tells everyone to adjust their mark according to the following formula:

    New mark = min(100, old mark/0.9),

    with all marks non-negative integers less than or equal to 100. What are the least upper and greatest lower bounds on the new average?

    How to Date a Mathie -- A Self Help Guide

    By Aurora Mendelsohn

    Due to an uncanny set of circumstances, I have dated three mathies, well maybe more, say, Pi mathies (What can I say -- I just love those pink ties). I can attribute my experience only to a depraved joke of God's or to strange mutations in my DNA. At any rate, I have learned much from my experience and I hope that others may benefit from my confusion. Mathies, read this! Hand it out to your boyfriends and girlfriends, and to your prospective dates. Better yet, simply pin it to a visible portion of your clothing as a warning label.


  • Carry paper at all times. If mathies don't have paper, then who knows what they'll write on.
  • Carry a compass and a map. (Need I say more...)
  • Familiarize yourself with the literature influential to molding the mathie's psyche: The Hitchhikers' Guide to the Galaxy trilogy (you can skip the fourth book in the series: the one about love), Monty Python's movies and Flying Circuses along with Star Trek movies and shows.
  • Learn some math buzz words and how they are grouped together. The actual mathematical concepts are irrelevant; the purpose of knowing this is to get the mathie's jokes. Here is a starter kit: {whole, natural, rational, irrational, integer}, {complex, imaginary, real, i}, {set, union, subset, intersection}.
  • Try to avoid references to the "real world." This will only confuse your mathie.
  • Go to parties and movies and other fun stuff. Mathies aren't nerds, contrary to popular belief.
  • Ask for help in your math courses. This is a prime fringe benefit of dating a mathie. Don't miss out! (However, when being helped never say, "I just want to know how to do it. Don't tell me what it means."
  • Learn how to play euchre (and bridge). This is probably what the mathie had in mind when s/he asked you if you wanted to be partners.
  • Make sure to inform your mathie that your idea of fine dining is not the Sidney Smith Cafeteria.
  • Remember where the car is parked. I suggest the method of carrying around a Polaroid camera and photographing the car and its relation to the parking lot.
  • When your mathie invites you out to a gathering of other mathies be prepared to not understand a single word that they are saying, even if they are all English words -- welcome to mathese. For example, all of the words in boldface in this article may be misinterpreted by a mathie.

    DO NOT

  • Announce "Puns are the lowest form of humour."
  • Go shopping for food, especially without a calculator! The mathie will ensure that you save that indispensible, extra 3.33 cents by buying the Super-JUMBO box of baking powder, even if it takes 3.33 hours to figure it out (and 3.33 years to use up that much).
  • Go out to dinner with a huge group of people and upon the arrival of the bill ask the mathie to figure out who owes what to whom. Otherwise, lots of money will change hands, the restaurant will be beginning to close and you will be further away from solving the problem of who owes what to whom than when you began. (In fact, to be on the safe side, do not even let the mathie see the bill! You don't need a manual check of the computer's or cash register's arithmetic abilities.)
  • Say, "Well, you are kind of cute, but I always thought engineers were better looking."
  • Allow the mathie to bring his/her laptop on a date with you. You + mathie = company (fun)! You + mathie + laptop = disaster date!
  • Get an e-mail account. If you have one, don't tell the mathie what it is. Otherwise, you may never receive a telephone call, letter or card from the mathie ever again. Somehow, "I love you" just doesn't have the same resonance on a computer screen.
  • Be impressed by your mathie's knowledge of Greek letters. Probability is, s/he doesn't speak a word of it.
  • Confuse the mathie's vocabulary with your own. HP does not mean steak sauce to a mathie, chips do not go with fish, nor does reading the news have anything to do with the Globe and Mail.
  • Ask a mathie, "So what are you going to do with a degree in math?" For some reason, this question tends to annoy them.
  • Ask the mathie to divide a cake or any other non-parallelogram shaped food. Assuredly all the pieces will be exactly even, but you won't want to serve guests bizarrely shaped creations.
  • Expect your mathie to add, subtract, multiply, divide, or count with any speed or accuracy. (If this really bothers you, go out with a nice, dependable HP calculator.)

    Good luck!

    About the cover:

    The donut is the 007 News' favourite snack; it seems to be the food of choice for undergraduate mathematics events at the Department of Mathematics. Not only that, the donut is the secret stereogram image hidden in the cover of Volume 8 Number 1, which was released in October 1994.

    For some strange reason, I could not locate the author of the image. Please identify yourself so credit can be given.

    Editor: Joel Chan
    Images scanned by Gary Farmaner, Holt Group, Department of Computer Science.

    MAT 007 I News is published from a corner of Sidney Smith Hall at the Department of Mathematics, University of Toronto. Issues are printed by the University of Toronto Press.

    Copyright © 1988-1990, 1992-1994, 1995 MAT 007 I News. All Rights Reserved.

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