130 // Paradox Lost|
Make all lengths
With the notation in the diagram, and bearing Pythagoras in mind, we have to find a, b and c so that all four of the numbers a2 þ b2 ; a2 þ c2 ; b2 þ c2 and a2 þ b2 þ c2 are perfect squares equal, respectively, to p2 ; q2 ; r2 and s2 . The existence of such numbers has neither been proved nor disproved, but some `near misses' have been found:
a ¼ 240; b ¼ 117; c ¼ 44; p ¼ 267; q ¼ 244; r ¼ 125;
but s is not an integer
a ¼ 672; b ¼ 153; c ¼ 104; q ¼ 680; r ¼ 185; s ¼ 697;
but p is not an integer
a ¼ 18;720; b ¼ 211;773;121; c ¼ 7;800; p ¼ 23;711;
q ¼ 20;280; r ¼ 16;511; s ¼ 24;961; but b is not an integer
If there is a perfect cuboid, it involves big numbers: it has been
........................................... proved that the smallest edge is at least 232 ¼ 4;294;967;296.
Paradox Lost In mathematical logic, a paradox is a self-contradictory statement the best known is `This sentence is a lie.' Another is Bertrand Russell's `barber paradox'. In a village there is a barber who shaves everyone who does not shave themselves. So who shaves the barber? Neither `the barber' nor `someone else' is logically acceptable. If it is the barber, then he shaves himself but we are told that he doesn't. But if it's someone else, then the barber does
When Will My MP3 Player Repeat? // 131|
not shave himself . . . but we are told that he shaves all such people, so he does shave himself.
In the real world, there are plenty of get-outs (are we talking about shaving beards here, or legs, or what? Is the barber a woman? Can such a barber actually exist anyway?) But in mathematics, a more carefully stated version of Russell's paradox ruined the life's work of Gottlob Frege, who attempted to base the whole of mathematics on set theory the study of collections of objects, and how these can be combined to form other collections.
Here's another famous (alleged) paradox:
Protagoras was a Greek lawyer who lived and taught in the fifth century BC. He had a student, and it was agreed that the student would pay for his teaching after he had won his first case. But the student didn't get any clients, and eventually Protagoras threatened to sue him. Protagoras reckoned that he would win either way: if the court upheld his case, the student would be required to pay up, but if Protagoras lost, then by their agreement the student would have to pay anyway. The student argued exactly the other way round: if Protagoras won, then by their agreement the student did not have to pay, but if Protagoras lost, the court would have ruled that the student did not have to pay.
Is this a genuine logical paradox or not?
Answer on page 279
When Will My MP3 Player Repeat? You have 1,000 songs on your MP3 player. If it plays songs `at random', how long would you expect to wait before the same song is repeated?
It all depends on what `at random' means. The leading MP3 player on the market `shuffles' songs just like someone shuffling a pack of cards. Once the list has been shuffled, all songs are 132 // When Will My MP3 Player Repeat?
played in order. If you don't reshuffle, it will take 1,001 songs to get a repeat. However, it is also possible to pick a song at random, and keep repeating this procedure without eliminating that song. If so, the same song might just come up twice in a row. I'll assume that all songs appear with the same probability, though some MP3 players bias the choice in favour of songs you play a lot.
You've probably met the same problem with birthdays replacing songs. If you ask people their birthday, one at a time, then on average how many do you have to ask to get a repeat? The answer is 23, remarkably small. There is a second, super- ficially similar problem: how many people should there be at a party so that the probability that at least two share a birthday is bigger than 1? Again the answer is 23. In both calculations we
2 ignore leap years and assume that any particular birthday occurs with probability 1/365. This isn't quite accurate, but it simplifies the sums. We also assume that all individuals have statistically independent birthdays, which would not be the case if, say, the party included twins.
I'll solve the second birthday problem, because the sums are easier to understand. The trick is to imagine the people entering the room one at a time, and to work out, at each stage, the probability that all birthdays so far are different. Subtract the result from 1 and you get the probability that at least two are equal. So we want to continue allowing people to enter until the probability that all birthdays are different drops below 1. 2
When the first person enters, the probability that their birthday is different from that of anyone else present is 1, because nobody else is present. I'll write that as the fraction
365 because it tells us that out of the 365 possible birthdays, all 365 have the required outcome.
When the second person enters, their birthday has to be When Will My MP3 Player Repeat? // 133
different, so there are now only 364 choices out of 365. So the probability we want is now
365 365 When the third person enters, they have only 363 choices, and the probability of no duplication so far is
365 364 363
365 365 365 The pattern should now be clear. After k people have entered, the probability that all k birthdays are distinct is
365 364 363 365 À k þ 1
6 6 6ÁÁÁ6
365 365 365 365 and we want the first k for which this is less than 1. Each fraction,
2 other than the first, is smaller than 1, so the probability decreases as k increases. Direct calculation shows that when k ¼ 22 the fraction equals 0.524 305, and when k is 23 it equals 0.492 703. So the required number of people is 23.
This number seems surprisingly small, which may be because we confuse the question with a different one: how many people do you have to ask for the probability that one of them has the same birthday as you to be bigger than 1? The answer to that is
2 much bigger in fact, it's 253.
The same calculation with 1,000 songs on an MP3 player shows that if each song is chosen at random then you have to play a mere 38 songs to make the probability of a repeat bigger than 1. The average number of songs you have to play to get a
2 repeat is 39 slightly more.
These sums are all very well, but they don't provide much insight. What if you had a million songs? It's a big sum a computer can do it, though. But is there a simpler answer? We can't expect an exact formula, but we ought to be able to find a
Professor Stewart's Cabinet of Mathematical Curiosities
Triangle of Cards
Turnip for the Books
The Four-Colour Theorem
Shaggy Dog Story
Shaggy Cat Story
Rabbits in the Hat
River Crossing 1 – Farm Produce
More Curious Calculations
Extracting the Cherry
Make Me a Pentagon
What is π?
Legislating the Value of π
If They Had Passed It …
Much Undo About Knotting
To Find Fake Coin
Mathematical Jokes 1
An Age-Old Old-Age Problem
Why Does Minus Times Minus Make Plus?
How to Unmake a Greek Cross
How to Remember a Round Number
The Bridges of Königsberg
How to do Lots of Mathematics
Euler's Pentagonal Holiday
Who Was Pythagoras?
Proofs of Pythagoras
A Constant Bore
Fermat's Last Theorem
A Little-Known Pythagorean Curiosity
Squaring the Square
Squares of Squares
Ring a-Ring a-Ringroad
Pure v. Applied
How Old Was Diophantus?
If You Thought Mathematicians Were Good at Arithmetic …
The Sphinx is a Reptile
Six Degrees of Separation
Duplicating the Cube
Curves of Constant Width
The Stolen Car
The Square Wheel
Why Can't I Divide by Zero?
River Crossing 2 – Marital Mistrust
Wherefore Art Thou Borromeo?
Kinds of People
The Sausage Conjecture
Tom Fool's Knot
What is the Golden Number?
What are the Fibonacci Numbers?
The Plastic Number
Don't Let Go!
The Most Likely Digit
Why Call It a Witch?
Möbius and His Band
Three More Quickies
Miles of Tiles
Why No Nobel for Maths?
Is There a Perfect Cuboid?
When Will My MP3 Player Repeat?
The Poincaré Conjecture
Pig on a Rope
The Surprise Examination
Mathematical Jokes 2
Why Gauss Became a Mathematician
What Shape is a Crescent Moon?
What is a Mersenne Prime?
The Goldbach Conjecture
Turtles All the Way Down
A Puzzling Dissection
A Really Puzzling Dissection
Nothing Up My Sleeve …
Nothing Down My Leg …
Can You Hear the Shape of a Drum?
What is e, and Why?
May Husband and Ay …
Many Knees, Many Seats
What Day is It?
Logical or Not?
A Question of Breeding
The Sixth Deadly Sin
How Deep is the Well?
What is the Square Root of Minus One?
The Most Beautiful Formula
Why is Euler's Beautiful Formula True?
Your Call May be Monitored for Training Purposes
Archimedes, You Old Fraud!
Fractals – The Geometry of Nature
The Missing Symbol
Where There's a Wall, There's a Way
Constants to 50 Places
Are Hard Problems Easy? or How to Win a Million Dollars by Proving the Obvious
Don't Get the Goat
All Triangles are Isosceles
If π isn't a Fraction, How Can You Calculate It?
Let Fate Decide
What Shape is a Rainbow?
The Riemann Hypothesis
Disproof of the Riemann Hypothesis
Murder in the Park
The Cube of Cheese
The Game of Life
Drawing an Ellipse – and More?
Mathematical Jokes 3
The Kepler Problem
The Milk Crate Problem
Spherical Sliced Bread
Professor Stewart's Cunning Crib Sheet
Share "Professor Stewart's Cabinet of Mathematical Curiosities":
Download for all devices (361 KB)