Mathematical mysteries
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Mathematical mysteries: Survival of the nicest?
http://plus.maths.org/content/mathematicalmysteriessurvivalnicest
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Helen Joyce </div>
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<p>One of the most puzzling aspects of human behaviour is cooperation, in situations where backstabbing and selfishness would seem to be more rewarding. From the point of view of evolutionary theory, the very existence of altruism and cooperation appear mysterious.</p>
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<div class="pub_date">Mar 2002</div>
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<h2>Survival of the nicest?</h2>
One of the most puzzling aspects of human behaviour is cooperation, in situations where backstabbing and selfishness would seem to be more rewarding. From the point of view of evolutionary theory, the very existence of altruism and cooperation appear mysterious. The mechanics of evolution seem to imply that rugged competition is the order of the day; that, given an opportunity to benefit by
cheating someone, or by defaulting on a deal, we will inevitably do so.<p><a href="http://plus.maths.org/content/mathematicalmysteriessurvivalnicest" target="_blank">read more</a></p>
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altruism
cooperation
evolution
game theory
Iterated Prisoners' Dilemma
Mathematical mysteries
Prisoner's Dilemma
Tit for Tat
Tit for Tat with forgiveness
Sat, 01 Dec 2001 00:00:00 +0000
plusadmin
4755 at http://plus.maths.org/content

Mathematical mysteries: The gentlemen from Basle and the Petersburg Paradox
http://plus.maths.org/content/mathematicalmysteriesgentlemenbasleandpetersburgparadox
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George Szpiro </div>
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<p>Just over 220 years have passed since the death of one of the most distinguished mathematicians in history: Daniel Bernoulli, who died on March 17th, 1782. The name of Bernoulli asks for precision since the family from Basle produced no fewer than eight outstanding mathematicians within three generations.</p>
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<div class="pub_date">November 2002</div>
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<h2>The difficult Bernoulli family</h2>
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<p>Daniel Bernoulli</p><p><a href="http://plus.maths.org/content/mathematicalmysteriesgentlemenbasleandpetersburgparadox" target="_blank">read more</a></p>
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22
bernoulli
coin games
expected prize
insurance
Mathematical mysteries
st. petersburg paradox
utility function
Fri, 01 Nov 2002 00:00:00 +0000
plusadmin
4759 at http://plus.maths.org/content

Mathematical mysteries: Transcendental meditation
http://plus.maths.org/content/mathematicalmysteriestranscendentalmeditation
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Helen Joyce </div>
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<p>Nineteenthcentury German mathematician Leopold Kronecker once said</p>
<p>God created the integers, all the rest is the work of man.</p>
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<div class="pub_date">September 2002</div>
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<p>Leopold Kronecker</p>
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<p>Nineteenthcentury German mathematician Leopold Kronecker once said</p><p><a href="http://plus.maths.org/content/mathematicalmysteriestranscendentalmeditation" target="_blank">read more</a></p>
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21
algebraic number
circlesquaring
e
irrational number
Mathematical mysteries
Pi
rational number
transcendental number
Sat, 31 Aug 2002 23:00:00 +0000
plusadmin
4758 at http://plus.maths.org/content

Mathematical mysteries: The Barber's Paradox
http://plus.maths.org/content/mathematicalmysteriesbarbersparadox
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Helen Joyce </div>
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<p>Suppose you walk past a barber's shop one day, and see a sign that says</p>
<p>"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."<br />
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you  does the barber shave himself?</p>
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<div class="pub_date">May 2002</div>
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<h2>A close shave for set theory</h2>
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<p>Suppose you walk past a barber's shop one day, and see a sign that says</p><p><a href="http://plus.maths.org/content/mathematicalmysteriesbarbersparadox" target="_blank">read more</a></p>
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20
Barber's Paradox
logic
Mathematical mysteries
philosophy of mathematics
Russell's Paradox
set theory
Theory of Types
what is impossible
ZermeloFraenkel axiomatisation of set theory
Tue, 30 Apr 2002 23:00:00 +0000
plusadmin
4757 at http://plus.maths.org/content

Mathematical mysteries: Strange Geometries
http://plus.maths.org/content/mathematicalmysteriesstrangegeometries
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Helen Joyce </div>
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<p>The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in  that is, to describe the geometric rules which govern it. Based on these axioms, he proved theorems  some of the earliest uses of proof in the history of mathematics.</p>
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<div class="pub_date">Jan 2002</div>
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<h2>Euclidean Geometry</h2>
<p>The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in  that is, to describe the geometric rules which govern it. Based on these axioms, he proved theorems  some of the earliest uses of proof in the history of mathematics. Euclid's work is discussed in detail in <a href="/issue7/features/proof1/index.html">The Origins
of Proof</a>, from Issue 7 of <i>Plus</i>.</p><p><a href="http://plus.maths.org/content/mathematicalmysteriesstrangegeometries" target="_blank">read more</a></p>
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18
curvature
curvature of space
escher
Euclid's Elements
Euclidean geometry
flatness
hyperbolic geometry
Mathematical mysteries
Mercator projection
spherical geometry
trigonometry
Sat, 01 Dec 2001 00:00:00 +0000
plusadmin
4754 at http://plus.maths.org/content

Mathematical mysteries: Painting the Plane
http://plus.maths.org/content/mathematicalmysteriespaintingplane
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Helen Joyce </div>
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<p>Suppose you have an infinitely large sheet of paper (mathematicians refer to this hypothetical object as the plane). You also have a number of different colours  pots of paint, perhaps. Your aim is to colour every point on the plane using the colours available. That is, each point must be assigned one colour.</p>
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<div class="pub_date">May 2001</div>
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<p>Suppose you have an infinitely large sheet of paper (mathematicians refer to this hypothetical object as the <i>plane</i>). You also have a number of different colours  pots of paint, perhaps. Your aim is to colour every point on the plane using the colours available. That is, each point must be assigned one colour.</p>
<p>Can you do this so that, for any two points on the plane which are exactly 1cm apart, they are given different colours?</p><p><a href="http://plus.maths.org/content/mathematicalmysteriespaintingplane" target="_blank">read more</a></p>
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Mathematical mysteries
plane colouring
Ramsey theory
Mon, 30 Apr 2001 23:00:00 +0000
plusadmin
4750 at http://plus.maths.org/content

Mathematical mysteries: Chomp
http://plus.maths.org/content/mathematicalmysterieschomp
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Helen Joyce </div>
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<p>Chomp is a simple twodimensional game, played as follows.<br />
Cookies are set out on a rectangular grid. The bottom left cookie is poisoned.<br />
Two players take it in turn to "chomp"  that is, to eat one of the remaining cookies, plus all the cookies above and to the right of that cookie.</p>
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<div class="pub_date">March 2001</div>
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<p>Chomp is a simple twodimensional game, played as follows.</p>
<p>Cookies are set out on a rectangular grid. The bottom left cookie is poisoned.</p>
<p align="center"><img src="/issue14/xfile/chomp.gif" /></p>
<p>Two players take it in turn to "chomp"  that is, to eat one of the remaining cookies, plus all the cookies above and to the right of that cookie.</p>
<p align="center"><img src="/issue14/xfile/cookiechomp.gif" /></p>
<p>The loser is the player who has to eat the poisoned cookie.</p><p><a href="http://plus.maths.org/content/mathematicalmysterieschomp" target="_blank">read more</a></p>
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Mathematical mysteries
strategy
Thu, 01 Mar 2001 00:00:00 +0000
plusadmin
4748 at http://plus.maths.org/content

Mathematical mysteries: Getting the most out of life  Part 1
http://plus.maths.org/content/mathematicalmysteriesgettingmostoutlifepart1
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Mark Wainwright </div>
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<p>There are many sorts of games played in a "bunco booth", where a trickster or sleightofhand expert tries to relieve you of your money by getting you to place bets  on which cup the ball is under, for instance, or where the queen of spades is. Lots of these games can be analysed using probability theory, and it soon becomes obvious that the games are tipped heavily in favour of the trickster!</p>
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<div class="pub_date">January 2001</div>
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<p>There are many sorts of games played in a "bunco booth", where a trickster or sleightofhand expert tries to relieve you of your money by getting you to place bets  on which cup the ball is under, for instance, or where the queen of spades is. Lots of these games can be analysed using probability theory, and it soon becomes obvious that the games are tipped heavily in favour of the
trickster! The punter is well advised to steer clear.<p><a href="http://plus.maths.org/content/mathematicalmysteriesgettingmostoutlifepart1" target="_blank">read more</a></p>
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Mathematical mysteries
Mon, 01 Jan 2001 00:00:00 +0000
plusadmin
4746 at http://plus.maths.org/content

Getting the most out of life  Part 2
http://plus.maths.org/content/gettingmostoutlifepart2
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Mark Wainwright </div>
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<p>The idea is this. To start with, you will choose an envelope at random, say by tossing a coin, and look at its contents, which is a cheque for some number  say n. (By randomising like this, you can be sure I haven't subconsciously induced you to prefer one envelope or the other.) You want to make sure that the bigger the number is, the more likely you are to keep it, in other words, the less likely you are to swap.</p>
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<div class="pub_date">January 2001</div>
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<p>The idea is this. To start with, you will choose an envelope at random, say by tossing a coin, and look at its contents, which is a cheque for some number  say <i>n</i>. (By randomising like this, you can be sure I haven't subconsciously induced you to prefer one envelope or the other.) You want to make sure that the bigger the number is, the more likely you are to keep it, in other
words, the less likely you are to swap.</p><p><a href="http://plus.maths.org/content/gettingmostoutlifepart2" target="_blank">read more</a></p>
http://plus.maths.org/content/gettingmostoutlifepart2#comments
13
Mathematical mysteries
Mon, 01 Jan 2001 00:00:00 +0000
plusadmin
4747 at http://plus.maths.org/content

Mathematical mysteries: Zeno's Paradoxes
http://plus.maths.org/content/mathematicalmysterieszenosparadoxes
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Rachel Thomas </div>
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<p>The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides.</p>
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<div class="pub_date">Nov 2001</div>
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<p>The paradoxes of the philosopher <a href="http://wwwgroups.dcs.standrews.ac.uk/~history/Mathematicians/Zeno_of_Elea.html">Zeno</a>, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the
philosophies of his teacher Parmenides.<p><a href="http://plus.maths.org/content/mathematicalmysterieszenosparadoxes" target="_blank">read more</a></p>
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17
Achilles Paradox
Arrow Paradox
convergence
geometric series
limit
Mathematical mysteries
relativity
worldline
Zeno's paradoxes
Fri, 01 Dec 2000 00:00:00 +0000
plusadmin
4753 at http://plus.maths.org/content

Mathematical mysteries: What colour is my hat?
http://plus.maths.org/content/mathematicalmysterieswhatcolourmyhat
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Mark Wainwright </div>
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<img class="imagefield imagefieldfield_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/30%20Jun%202010%20%2017%3A33/mystery2.gif?1277915590" /> </div>
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<p>This is a game played between a team of 3 people (Ann, Bob and Chris, say), and a TV game show host. The team enters the room, and the host places a hat on each of their heads. Each hat is either red or blue at random (the host tosses a coin for each teammember to decide which colour of hat to give them). The players can see each others' hats, but noone can see their own hat.</p>
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<div class="pub_date">Sep 2001</div>
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<p>This is a game played between a team of 3 people (Ann, Bob and Chris, say), and a TV game show host. The team enters the room, and the host places a hat on each of their heads. Each hat is either red or blue at random (the host tosses a coin for each teammember to decide which colour of hat to give them). The players can see each others' hats, but noone can see their own hat.</p><p><a href="http://plus.maths.org/content/mathematicalmysterieswhatcolourmyhat" target="_blank">read more</a></p>
http://plus.maths.org/content/mathematicalmysterieswhatcolourmyhat#comments
16
errorcorrecting code
game theory
Hamming code
Mathematical mysteries
strategy
Fri, 01 Dec 2000 00:00:00 +0000
plusadmin
4752 at http://plus.maths.org/content

Mathematical mysteries: The Solitaire Advance
http://plus.maths.org/content/mathematicalmysteriessolitaireadvance
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<p>Solitaire is a game played with pegs in a rectangular grid. A peg may jump horizontally or vertically, but not diagonally, over a peg in an adjacent square into a vacant square immediately beyond. The peg which was jumped over is then removed.</p>
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<div class="pub_date">September 2000</div>
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<p>Solitaire is a game played with pegs in a rectangular grid. A peg may jump horizontally or vertically, but not diagonally, over a peg in an adjacent square into a vacant square immediately beyond. The peg which was jumped over is then removed.</p><p><a href="http://plus.maths.org/content/mathematicalmysteriessolitaireadvance" target="_blank">read more</a></p>
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12
Mathematical mysteries
Thu, 31 Aug 2000 23:00:00 +0000
plusadmin
4745 at http://plus.maths.org/content

Mathematical mysteries: Right angle race
http://plus.maths.org/content/mathematicalmysteriesrightanglerace
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<p>The German mathematician Adam Ries (14921559) was the author of the most successful textbook of commercial arithmetic of his day. The book, published in 1552, earned such a high reputation that the German phrase nach Adam Ries is used to this day to indicate a correct calculation.</p>
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<div class="pub_date">June 2000</div>
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<p>The German mathematician <a href="http://wwwgroups.dcs.stand.ac.uk/~history/Mathematicians/Ries.html">Adam Ries</a> (14921559) was the author of the most successful textbook of commercial arithmetic of his day. The book, published in 1552, earned such a high reputation that the German phrase <em>nach Adam Ries</em> is used to this day to indicate a correct calculation.</p><p><a href="http://plus.maths.org/content/mathematicalmysteriesrightanglerace" target="_blank">read more</a></p>
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11
Mathematical mysteries
Wed, 31 May 2000 23:00:00 +0000
plusadmin
4744 at http://plus.maths.org/content

Mathematical mysteries: Foucault's pendulum and the eclipse
http://plus.maths.org/content/mathematicalmysteriesfoucaultspendulumandeclipse
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<p>You may have seen Foucault's pendulum. There's one in the Science Museum in London (part of the National Museum of Science and Industry), and there are many more in various locations around the UK (for instance, in Glasgow) and the world (including one at the United Nations Headquarters and a famous example at Le PanthÃ©on in Paris).</p>
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<div class="pub_date">September 1999</div>
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<p>You may have seen Foucault's pendulum. There's one in the <a href="http://www.sciencemuseum.org.uk">Science Museum</a> in London (part of the <a href="http://www.nmsi.ac.uk/">National Museum of Science and Industry</a>), and there are many more in various locations around the UK (for instance, in Glasgow) and the world (including one at the <a href="http://www.un.org">United Nations
Headquarters</a> and a famous example at <a href="http://www.paris.org/Monuments/Pantheon/">Le PanthÃ©on</a> in Paris).</p><p><a href="http://plus.maths.org/content/mathematicalmysteriesfoucaultspendulumandeclipse" target="_blank">read more</a></p>
http://plus.maths.org/content/mathematicalmysteriesfoucaultspendulumandeclipse#comments
9
Mathematical mysteries
Tue, 31 Aug 1999 23:00:00 +0000
plusadmin
4766 at http://plus.maths.org/content

Mathematical mysteries: How unilluminating!
http://plus.maths.org/content/mathematicalmysterieshowunilluminating
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<p>In the 1950's, Ernst Straus asked a seemingly simple problem. Imagine a dark room with lots of turns and sidepassages, where all the walls are covered in mirrors  just like the Hall of Mirrors in an oldfashioned funfair. Is it true that if someone lights a match somewhere in the room, then wherever you stand in the rest of the room (even down a sidepassage) you can see a reflection of the match?</p>
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<div class="pub_date">May 1999</div>
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<p>In the 1950's, Ernst Straus asked a seemingly simple problem. Imagine a dark room with lots of turns and sidepassages, where all the walls are covered in mirrors  just like the Hall of Mirrors in an oldfashioned funfair. Is it true that if someone lights a match somewhere in the room, then wherever you stand in the rest of the room (even down a sidepassage) you can see a reflection of the
match?</p><p><a href="http://plus.maths.org/content/mathematicalmysterieshowunilluminating" target="_blank">read more</a></p>
http://plus.maths.org/content/mathematicalmysterieshowunilluminating#comments
8
Mathematical mysteries
Fri, 30 Apr 1999 23:00:00 +0000
plusadmin
4765 at http://plus.maths.org/content