Gotcha: Paradoxes to Puzzle and Delight W. 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.

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Gotcha: Paradoxes to Puzzle and Delight W. 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.

Cretans are detestable, disobedient and unfit for doing anything good. 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. Sequel: There was a young man of Verdun. Make each pronoun agree with their antecedent When dangling, watch your participles. Verbs has to agree with their subjects.

About those sentence fragments. Try to not ever split infinitives. Always read what you have written to see you any words out Correct spelling is esential.

Eschew obfuscation. Alfred Marshall, British Economist: "Every short sentence about economics is inherently false. Hochschild is mentioned nowhere else, except in this entry, which is on p. Kalin: It went into an oscillating phase, making "a hell of a racket". 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.

Circular Paradoxes 12 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. Grelling: two sets of adjectives: self-descriptive e. 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? Paradoxes and metalanguage Semantic paradoxes : depend on truth-value; e. All semantic paradoxes can be translated into set-theory paradoxes, e. But then what it asserts is true and it cannot. 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.

The sum of the interior triangles are deg B. 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. You write "YES" if you predict that the event will happen.

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. Number paradoxes 33 Classes of numbers have started as paradoxes that violate some intuition: 1.

The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. 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, Exactly what addition means will be defined later on: just consider them as names. 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.

He tests many people on this: Here are two boxes, A and B. 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. 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. 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. 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.

One can make the argument for taking both boxes even more vivid by changing the setup a bit. The moderator is watching you decide between one box and both boxes, and the money is there in front of his eyes. 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. 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. Yet none of them has been completely convincing, so the debate goes on. What is most likely? 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. Now shuffle - and holding the pack under the table, say you can "feel" colour differences - and produce a pair of colours every time.

Now deal out 26 cards reversing the order and shuffle. 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! Average Median Mode Mr. Gizmo hires Sam. We pay very well. Avg salary is Yours is , but will increase.

After a few days Sam finds that everyone around is earning , and challenges Mr Gizmo. But Sam would have been more interested in the median or even perhaps in a "typical" salary, or the "mode".

No avg family. 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. Maybe mode - e. Can apply to how the same woman rates three men A, B, C, on intelligence, good looks, and income.

Relations like "taller than", "bigger than", "less than", "equals", heavier than, etc. The voting puzzle boggles the mind because one expects "prefers" to be transitive.

Also called the Arrow paradox after economist Kenneth Arrow. 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. What speed must he come down at so his avg speed is 10 kph? 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?

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## Aha! Gotcha: Paradoxes to Puzzle & Delight (Tools for Transformation)

The word paradox has many meanings, but I use it here in a broad sense to include any result so contrary to common sense and intuition that it invokes an immediate emotion of surprise. Such paradoxes are of four main types: An assertion that seems false but actually is true. An assertion that seems true but actually is false. A line of reasoning that seems impeccable but which leads to a logical contradiction.

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