Gardner, Martin;
Aha! Gotcha: Paradoxes to Puzzle and Delight
W.H. Freeman & Company 1982-04 [gbook]
ISBN 9780716713616 / 0716713616 paperback
topics: | math | puzzle
Aha! Gotcha is my top mathematical puzzle book of all time. An amazing selection of puzzles, set in a mathematical background that is accessible from middle school on - and yet, all the problems have immense depth, and can undergo intense explorations. My sons, Zubin and Zagreb, really loved the book - Zubin devouring it when he was in grade 6. Every page you turn to has something new, even if you have considerable exposure to mathematics.
Bible: Titus 1:12-13 (New Intl Version) 12 Even one of their own prophets has said, "Cretans are always liars, evil brutes, lazy gluttons." 13 This testimony is true. - St Paul's epistle to Titus [In the context of why Titus was left in Crete, to handle the "rebellious people, mere talkers and deceivers, especially those of the circumcision group.... Cretans are detestable, disobedient and unfit for doing anything good."; not clear if Paul was aware of the paradox. ] Stoic philosopher Chrysippus wrote 6 treatises on liar paradox (all lost) Poet Philetus of Cos worried himself so thin that he carried lead in his shoes to prevent being blown away, and eventually died an early death
short story: "Told under oath" by Lord Dunsany man pledges under solemn oath that the story he is about to tell is the whole truth and nothing but.... he had met Satan at a party, and the two struck a bargain, so that the man, who was the worst golfer in the club, would now always make a hole in one. After repeated holes in one, everybody became convinced that the man was somehow cheating, and he was expelled from the club. When Dunsany asks him what Satan got in return for his gift, he says that Satan had "extorted from me my power of ever speaking the truth again." Graffitti on wall: Down with Graffitti! (IDEA: Office memo in Kanpur: "Ban Office Memos") Limerick paradox: There was a lady of Crewe Whose limericks stopped at line two. Sequel: There was a young man of Verdun. (why does this work? mind conjures up second line...)
Harold Evans, on page 182 of his 1972 book Newsman's English, attributes
these "un-rules" to Helen Ferril of the Rocky Mountain News.
Don't use no double negatives.
Make each pronoun agree with their antecedent
When dangling, watch your participles.
Don't use commas, which aren't necessary.
Verbs has to agree with their subjects.
About those sentence fragments.
Try to not ever split infinitives.
It is important to use your apostrophe's correctly.
Always read what you have written to see you any words out
Correct spelling is esential.
Other similar ones:
http://www.math.tamu.edu/~boas/courses/math696/rools.html
The passive voice is to be avoided.
Eschew obfuscation.
Don't abbrev.
Alfred Marshall, British Economist: "Every short sentence about economics is
inherently false."
Paul R Halmos, "Finite Dim Vector Spaces,"
has a reference to : Hochschild, G.P. 198. Hochschild is mentioned nowhere
else, except in this entry, which is on p. 198.
Raymond Smullyan, "This book needs no title"
Clearly the following is false:
"This sentence contains seven words"
Then it's opposite must be true?
"This sentence does not contain seven words"
In 1947, William Burkhart and Theodore Kalin, UGs at Harvard, programmed a
computer to evaluate a version of the liar paradox. Kalin: It went into an
oscillating phase, making "a hell of a racket". 9
Infinite regress: Which came first: The chicken or the egg?"
Visual regress: cover of book in cover of book... (e.g. russell/norvig, AIMA)
Jonathan Swift poem, reworked by logician Augustus de Morgan in "A Budget of
Paradoxes":
Great fleas have little fleas
Upon their backs to bite 'em
And little fleas have lesser fleas,
and so on ad infinitum.
And the great fleas, themselves, in turn,
Have greater fleas to go on;
While these again have greater still,
And greater still, and so on.
Plato: The next statement of Socrates will be false. Socrates: Plato has spoken truly! On one side of a card: The sentence on the other side of this card is true. On other side: The sentence on the other side of this card is false. Chapter 4 of Through the Looking Glass: The King is asleep. Tweedledee tells Alice that the King is dreaming about her, and that she has no existence except as a "sort of thing" in the king's dream. But the entire dialogue occurs in Alice's own dream. http://www.sabian.org/Alice/lgchap04.htm: "Are there any lions or tigers about here?" she asked timidly. "It's only the Red King snoring," said Tweedledee. "Come and look at him!" the brothers cried, and they each took one of Alice's hands, and led her up to where the King was sleeping.... "He's dreaming now," said Tweedledee: "and what do you think he's dreaming about?" Alice said "Nobody can guess that." "Why, about you!" Tweedledee exclaimed, clapping his hands triumphantly. "And if he left off dreaming about you, where do you suppose you'd be?" "Where I am now, of course,"said Alice. "Not you!" Tweedledee retorted contemptuously. "You'd be nowhere. Why, you're only a sort of thing in his dream!" "If that there King was to wake," added Tweedledum, "you'd go out-- bang!--just like a candle!" "I shouldn't!" Alice exclaimed indignantly. "Besides, if I'm only a sort of thing in his dream, what are you, I should like to know?" "Ditto," said Tweedledum. "Ditto, ditto!" cried Tweedledee. He shouted this so loud that Alice couldn't help saying "Hush! You'll be waking him, I'm afraid, if you make so much noise." --> visual analogy: escher "Drawing Hands" - 2 hands drawing themselves [see Godel Escher Bach] p.17: Russell's Barber paradox: I shave all men in town, and only those men, who do not shave themselves. Grelling: two sets of adjectives: self-descriptive (e.g. "short", "polysyllabic"), and non-self-descriptive (e.g. "monosyllabic", "long", "Bengali") etc. To which class does "non-self-descriptive" belong? Max Black: consider integers mentioned in this book. Fix your attention on the smallest integer not mentioned to in any way in the book. Can this be done?
Semantic paradoxes : depend on truth-value; e.g. all cretans are liars Set-theory paradoxes: depend on sets - typically infinite sets. All semantic paradoxes can be translated into set-theory paradoxes, e.g. this sentence is false = This assertion is a member in the set of all false assertions. But then what it asserts is true and it cannot. and so on. Tarski: Handling paradoxes in "metalanguages" - as opp to "object language"; metalanguage includes all of object language, and also about the truth values about statements in the obj lg. T/F-hood of metalanguage statements can only be handled in a meta-meta-language and so forth in an infinite ladder. 21 e.g. A. The sum of the interior triangles are 180 deg B. sentence A is true C. sentence B is true D. sentence C is true Language at level A is about theorems in geometry. Language at level B is in geometry textbooks etc. Books about proof theory are written in lg C; seldom need to go beyond level C. Russell: Theory of types - not permissible to say that a set is a member of itself, or not a member of itself. 23 [Analog: Thus sentences at diff meta-levels are not permitted. ]
Swami (astrologer) and daughter: Daughter: I have written an event on a card. You write "YES" if you predict that the event will happen. You write "NO" if you predict it won't happen. If you are correct, you don't have to buy me a car now. Else you buy my graduation car now. She has written on the card "You will write NO". Whether the Swami writes Yes or No, he will be false.
Classes of numbers have started as paradoxes that violate some intuition:
1. irrational numbers
2. imaginary
3. violate commutativity, e.g. quaternions
4. violate associativity, e.g. cayley numbers
5. transfinite or infinite numbers, aleph-1, aleph-2
AXIOM OF CHOICE in set theory: says that given any collection of bins, each
containing at least one object, it is possible to make a selection of exactly
one object from each bin, even if there are infinitely many bins and there is
no "rule" for which object to pick from each. The axiom of choice is not
required if the number of bins is finite or if such a selection "rule" is
available.
ORDINAL NUMBER
A natural number (including 0) can be used for two purposes: to describe the
size of a set, or to
describe the position of an element in a sequence. When restricted to finite
sets these two concepts coincide; there is only one way to put a finite set
into a linear sequence, up to isomorphism. When dealing with infinite sets
one has to distinguish between the notion of size, which leads to cardinal
numbers, and the notion of position, which is generalized by the ordinal
numbers described here. This is because, while any set has only one size (its
cardinality), there are many nonisomorphic well-orderings of any infinite
set.
Example: start with the natural numbers, 0, 1, 2, 3, 4, 5, ... After all
natural numbers comes the first infinite ordinal, ω, and after that come ω+1,
ω+2, ω+3, and so on. (Exactly what addition means will be defined later on:
just consider them as names.) After all of these come ω·2 (which is ω+ω),
ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of
ordinals which we form in this way (the ω·m+n, where m and n are natural
numbers) must itself have an ordinal associated to it: and that is
ω². Further on, there will be ω3, then ω4, and so on, and ω^ω, then ω^ω², and
much later on ε0 (epsilon nought) (to give a few examples of relatively small
—countable — ordinals). We can go on in this way indefinitely far
("indefinitely far" is exactly what ordinals are good at: basically every
time one says "and so on" when enumerating ordinals, it defines a larger
ordinal). The smallest uncountable ordinal is the set of all countable
ordinals, expressed as ω1.
Alien creature [or a genie] Omega is able to predict what humans will choose,
and appears to be widely successful.
He tests many people on this:
Here are two boxes, A and B. A is transparent and has 1K $. B is opaque; it
may or may not contain 1mn $.
You may choose to take both boxes. But if I expected you to do this, I have
left box B empty. On the other hand, you may take only box B. If I expected
you to do this, I have put 1 million $ in it.
A man and a woman have diff views:
Man: So far in all the tests, Omega has been accurate. So let me take B, and
get the 1mn.
Woman: Whatever he has done, is done. Now if I take only B, it may be empty
and I get nothing, whereas picking both I get 1K. If B has 1mn, even
then, picking both I am richer by 1K.
This paradox is related to the belief in free will : can the future be fully
determined? If you believe in free will, you take both; if you believe in
determinism, you take only B. [See article by Robert Nozick in the Sci Am,
Jul 1973]
(from http://members.aol.com/kiekeben/newcomb.html) As Nozick points out, there are two accepted principles of decision theory in conflict here. The expected-utility principle (based on the probability of each outcome) argues that you should take the closed box only. The dominance principle, however, says that if one strategy is always better, no matter what the circumstances, then you should pick it. And no matter what the closed box contains, you are $1000 richer if you take both boxes than if you take the closed one only. One can make the argument for taking both boxes even more vivid by changing the setup a bit. For instance, suppose that the closed box is open on the face opposing you, so that you can't see its contents but an experiment moderator can. The moderator is watching you decide between one box and both boxes, and the money is there in front of his eyes. Wouldn't he think you are a fool for not taking both boxes? from http://www.slate.com/?id=2061419: "I have put this problem to a large number of people, both friends and students in class," Nozick wrote in the 1969 article. "To almost everyone it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly." When Martin Gardner presented Newcomb's Problem in 1973 in his Scientific American column, the enormous volume of mail it elicited ran in favor of the one-box solution by a 5-to-2 ratio. (Among the correspondents was Isaac Asimov, who perversely plumped for the two-box choice as an assertion of his free will and a snub to the predictor, whom he identified with God.) Newcomb, the begetter, was a one-boxer. Nozick himself started out as a lukewarm two-boxer, despite being urged by the decision theorists Maya Bar-Hillel and Avishai Margalit to "join the millionaires' club" of one-boxers. By the '90s, however, Nozick had arrived at the unhelpful view that both arguments should be given some weight in deciding which action to take. After all, he reasoned, even resolute two-boxers will become one-boxers if the amount in Box A is reduced to $1, and all but the most diehard one-boxers will become two-boxers if it is raised to $900,000, so nobody is completely confident in either argument. The quantity and ingenuity of the resolutions proposed for Newcomb's Problem over the years have been staggering. (It has been linked to Schrödinger's Cat in quantum mechanics and Maxwell's Demon in thermodynamics; more obviously, it is analogous to the Prisoner's Dilemma, where the other prisoner is your identical twin who will almost certainly make the same choice you do to cooperate or defect.) Yet none of them has been completely convincing, so the debate goes on. Could Newcomb's Problem turn out to have the longevity of Zeno's paradoxes? Will philosophers still be vexing over it 2,500 years from now, long after [Nozick's] Anarchy, State, and Utopia is forgotten?
Cats have litter of 4. How many boys / girls? What is most likely? 4B or 4G is unlikely (1/8). 2B 2G? Turns out 2-2 is less likely (3/8) than a 1-3 split (1/2). In bridge, a 4-3-3-3 hand has a chance of 1/8-1/9. A 4-4-3-2 hand is 1/5, and a 5-3-3-2 is 1/6 --- Shuffling Card tricks (must be exact shuffle): [Gilbreath principle] 126 originally, arrange cards so that colours are alternating. Now cut so bottom colours are different. Now shuffle thoroughly. Pick up top two cards - will always alternate in colour. Presentation - have cards pre-arranged. Then have friend deal out 26 cards (this reverses the order, and ensures bottom cards don't match). Now shuffle - and holding the pack under the table, say you can "feel" colour differences - and produce a pair of colours every time. Same trick with suits - pre-arrange so it is SHCDSHCD etc. Now deal out 26 cards (reversing the order) and shuffle. Now the resulting pile will have SHCD SHCD etc, so top 4 cards are from each suit. Two packs: Arrange in same order. Deal 52 cards off the top (reversing). Shuffle the two decks - take top 52 cards - it is a complete deck!
Mr. Gizmo hires Sam. We pay very well. Avg salary is 600. Yours is 150, but will increase. After a few days Sam finds that everyone around is earning 200, and challenges Mr Gizmo. The breakup is : Gizmo: 4800, his bro: 2000, six relatives on board: 500, 5 foremen: 400, ten workers: 200 - total 13,800 / 23 = 600. But Sam would have been more interested in the median (400) or even perhaps in a "typical" salary, or the "mode". "Average" num of children / family in Russia = 1.25. No avg family. But "typical" family = mode - exists. (or height, or xxx) is meaningless - may be 1.5 children. Median always exists if # samples odd, or if two median values equal. Else taken to be mean of the two central values, so will not exist. Modes - may be more than one, if two peaks. CATEGORY: PROTOTPES = MEAN or MEDIAN or MODE? (Maybe mode - e.g. some colour names have two prototypes - two modes)
Statistics show that more people die of TB in Arizona than in any other state. So does Arizona's climate favour TB? If the opp. holds, AZ is good for TB - then maybe more TB patients come to AZ to convalesce, so of course more die... Statistical conclusions have nothing to do with cause and effect. e.g.: Statistics show that most (say 80%) car accidents occur when travelling at moderate speeds (say < 40kmph), and that very few accidents (say 1%) occur while going at 150kmph. So is it safer to drive at 150kph? e.g. Recent study showed that most mathematicians were eldest sons. So do elder sons have a > ability? "100 families each have two children". What fraction of the boys will be eldest sons? Answer: 3/4 [Same is true for daughters]. [eldest son: older among sons - so in GB also B is eldest son; except each family: BG BB GB GG: equi-probability events - eldest son in 3/4] Modern advertising (e.g. tv) often replete w such misleading statistics.
Four people meet. Odds that two have the same birthsign > 0.4 (1 - 11/12 x 10/12 x 9/12 = 1- 55/96 = 42/96) Five people in group. Chances that birthdays in same month = 89/144 = 0.62 Four people, chances b'days on same day of week = 1 - 120/343 = 0.65 K people, chances b'days are identical? p>0.5 if K > 23
each thirds of voters 1: A > B > C 2: B > C > A 3: C > A > B 2/3ds of the voters prefer A to B. 2/3ds prefer B to C. But most do not prefer A to C; in fact, 2/3ds prefer C to A. Can apply to how the same woman rates three men A, B, C, on intelligence, good looks, and income. Taken by pairs, she may find A>B, B>C, C>A. old puzzle, going back to 18th c., deals with the situation where pairwise choices do not match up. Relations like "taller than", "bigger than", "less than", "equals", heavier than, etc. are transitive, so that if xRy AND yRz then xRz. The voting puzzle boggles the mind because one expects "prefers" to be transitive. Also called the Arrow paradox after economist Kenneth Arrow. Also known as Arrow's impossibility theorem, he proved it in his Phd thesis and later in the book "Social Choice and Individual Values" (1951). The theorem states that no voting system with three or more discrete options to choose from can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of reasonable criteria. Restaurant serving apple, blueberry and cherry pie. Any day, only two are served. They may find A>B, B>C, C>A. [Halmos] 129 Richard G Niemi/William H Riker: Choice of Voting Systems June 1976, Sci Am Martin Gardner column, Oct 1974 Lynn Steen, Math Games column Oct 1980 --- 142 Computing Averages-- skier going on uphill lift at 5 kmph. What speed must he come down at so his avg speed is 10 kph? [Ans: infinite] --- 145 Worm on elastic rope-- Worm on rubber rope, travels 1cm/second. Rope is initially 1km long. After 1 second, it stretches to 2km, next second, it stretches to 3km, etc. Will the worm ever reach the end? [Hint: Think of rope being initially 2cm long] [Hint2: worm also moves when rope expands.] In 1st second, w moves 1/100K length. Then it goes to 2km, and then it moves 1/200K and so on. So after k seconds, it has moved: 1/100K [ 1 + 1/2 + 1/3 + 1/4 + ... 1/k] This is the harmonic series. It does not converge (every 2^k terms is greater than the k/2). The worm will reach the end in about e^100k terms, far more than age of universe. by then rubber band will be molecules separated by vast spaces.
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