Logic and Reasoning

A Bogus Proof

Here is a clear demonstration of a startling result.
Theorem: 1 = 2

Proof: Let a and b be integers, with a = b. Then ab = b^2. Thus,
ab - b^2 = a^2 - b^2
and b(a - b) = (a + b)(a - b),
whence b = a + b,
so that b = 2b.
Therefore, 1 = 2.

Is there a flaw in this proof?

Please show me the solution -- I give up

Who owns the Zebra?

The following puzzle is a fine example of what I call “detective puzzles:” Based on clues supplied in a narrative, one is to answer a question by applying simple, man-on-the-street logic to the information (not all of it relevant) supplied.

On an odd little street in the town of Somewhere, there are five house in a row. Each house is a different colour, each is inhabited by a woman of different nationality, and the owner of the houses also have their differences: each owner has a different pet, prefers a different drink and works in a different profession. A detective, charged with the task of discovering who drinks water and who owns the Zebra, gathered the following information, itemized for your convenience:

1. The Englishwoman lives in the red house.
2. The Spaniard owns a dog.
3. Coffee is drunk in the green house
4. The Ukrainian drinks tea.
5. The green house is immediately to the right of the Ivory house.
6. The engineer owns the snail.
7. The diplomat lives in the yellow house.
8. Milk is drunk in the middle house.
9. The Norwegian lives in the first house on the left.
10. The doctor lives next to the owner of the fox.
11. The diplomat lives next to the owner of the horse.
12. The teacher drinks orange juice.
13. The carpenter is Japanese.
14. The Norwegian lives next to the blue house.

Please show me the solution -- I give up

Achilles and the Tortoise

Achilles, fleet of foot, is to race a Tortoise who is ponderously slow. In fairness, Achilles gives the Tortoise a handicap, say 100 metres. The course of the race is one kilometre long. The Tortoise begins as the starting pistol fires (never mind the anachronism) and beetles ponderously along until it reaches the 100-metre mark, at which time Achilles is off like the wind. But wait! At every moment the Tortoise is in one place and Achilles in another. Thus, as the race progresses, Achilles has never been in more places than the turtle has. It follows that Achilles can never catch up to his competitor.

Please show me the solution -- I give up

The Bee and the Trains

Owing to a mixup at the control centre no doubt, two trains begin a journey from two different cities, Centerville and Browntown, toward each other along the same set of tracks. The towns are 100 km apart and the trains each travel at 50 km per hour.

If this situation were not strange enough already, a bee sits on the headlamp of the Centerville train before the trip begins. At the instant the trains set off, the bee flies straight down the track toward the Browntown train, rebounds from its headlamp and heads back along the track toward the Centerville train. When it meets that train, it rebounds again, only to head back to the Browntown train. On and on the bee goes at a constant 60 km per hour, rebounding from train to train, until the two collide. What is the total distance traveled by the bee before the disaster takes place?

Please show me the solution -- I give up

Word Magic

Invented by Lewis Carroll, this class of puzzles involves the transformation of one word into another by single-letter letter substitutions. The result is a form of magic, in which one thing is changed into another. For example, one can change a "cup" to a "rib" by the following sequence of substitutions: cup --> cap --> rap --> rip --> rib. Itıs not too difficult to change a "dog" into a "cat," but can you change "snow" into "rain?" Of course, every intervening word must be in your favorite dictionary.

Please show me the solution -- I give up

The Monkey and the Coconuts

Five men and a monkey were on an ocean voyage (with no obvious aim in mind) when a fierce storm came up and their boat sank, right next to a desert island, as luck would have it. The island was plentifully supplied with coconuts, fortunately, and they spent the first day gathering the coconuts, heaping them in a great pile beside their camp.

That night, one of the men was sleepless at the thought that everyone would fight over the coconuts at daybreak. So he arose, crept to the pile, and counted out the coconuts. Except for one coconut, it was possible to divide the pile into five equal portions. When he felt a tug on his coat, he recognized the monkey in the darkness. He gave the extra coconut to the monkey, buried his share of the nuts, then went back to a sound sleep.

Then another man got up and stole to the pile with the same object in mind. Again there was one coconut left over and again the monkey made its presence known. The second man gave the spare coconut to the monkey, buried (what he thought was) his fair share, and went back to sleep.

One by one, the remaining men got up and did the same. In the morning, the men divided what coconuts remained into five equal piles, each taking his share and saying nothing about the night just past. Oh, yes. There was again one coconut left over and the men cheerfully gave it to the monkey.

How many coconuts did the five men and the monkey assemble on the previous day?

Please show me the solution -- I give up

The Philosophers and the Owls

One day three Greek philosophers settled under the shade of an olive tree, opened a bottle of Retsina, and began a lengthy discussion of the Fundamental Ontological Question: Why does anything exist? After a while, they began to ramble. Then, one by one, they fell asleep. While the men slept, three owls, one above each philosopher, completed their digestive process, dropped a present on each philosopher's forehead, the flew off with a noisy "hoot."

Perhaps the hoot awakened the philosophers. As soon as they looked at each other, all three began, simultaneously, to laugh. Then, one of them abruptly stopped laughing. Why?

Please show me the solution -- I give up

The Unexpected Hanging

The Queen of Quantania has imprisoned the wily thief, Qualman. She has sentenced him to die unexpectedly on one of the seven days following his internment. On the morning of each day, the Grand Vizier will appear before Qualman's cell and on exactly one of those visits, he will be followed, five minutes later, by the royal executioner.

Perhaps because she has a soft spot for Qualman or perhaps because she is a sporting type, the Queen tells Qualman that if he correctly predicts the day of his hanging to the Grand Vizier at the moment he arrives on the fateful morning, she will commute his sentence to a year of service in the Royal baths.

Qualman replies. "Then you can never hang me. You may as well send me to the baths directly. "How so?" says the Queen. "Well, you cannot execute me on the last day, for I will surely predict that day, when it comes." "Very well," says the Queen. "And your highness cannot execute me on the second last day because when that day comes, I will already know that it cannot be on the morrow, so it must be that day."

"I see," said the Queen. "And so on?"

"And so on," replied Qualman.

And yet, what if Qualman's executioner shows up on the third day, say?

Please show me the solution -- I give up

The Beans and the Pot

A pot contains 75 white beans and 150 black ones. Next to the pot is a large pile of black beans. A somewhat demented cook removes the beans from the pot, one at a time, according to the following strange rule: He removes two beans from the pot at random. If at least one of the beans is black, he places it on the bean-pile and drops the other bean, no matter what colour, back in the pot. If both beans are white, on the other hand, he discards both of them and removes one black bean from the pile and drops it in the pot.

At each turn of this procedure, the pot has one less bean in it. Eventually, just one bean is left in the pot. What colour is it?

Please show me the solution -- I give up

The Sinking Ship

Thirty people elect to cross the sea in an unsafe ship. Half the people belong to social/ethnic group A, half to B. (Historical version of this puzzle have used Christians and Turks, Blacks and Whites, Sluggards and Scholars) A storm comes up and the boat springs a leak which cannot be controlled unless half the people jump overboard. Someone from group A suggests that everyone get in a circle, which, to be fair, the proposer "randomizes." Then everyone agrees that, starting at a specified person, every ninth passenger will jump overboard. Strangely, everyone from group B ends up jumping overboard. How did the proposer manage it?

Please show me the solution -- I give up

The Seven Sevens

In the following division question all digits but seven have been replaced by asterisks.

            **7**

****7*)**7*******

       ******

       ------

       *****7*

       *******

       -------

         *7****

         *7****

         ------

         *******

         ****7**

         -------

           ******

           ******

           ------

Knowing that the division works out evenly, with **7** the answer and no remainder, use deduction to infer the missing digits.

Please show me the solution -- I give up

Spies versus Spies

Two spies, John and Bill (not their real names, obviously) work for the government of Kryptopia by ferreting out moles infiltrating from the country of Klandestein. At present, there is just one mole from Klandestein in Kryptonia, but John and Bill have each been investigating eight suspects. They have finally narrowed their respective lists of suspects to just two out of the original eight. If their short lists overlap in one suspect, they will pick him up for "questioning".

Of course, agents from Klandestein have phone taps everywhere, so John and Bill must devise a method for determining the common suspect without giving this information over the phone. Can you devise a method by which the two agents can communicate over the telephone so that, in the end, each knows the common suspect but the telephone interloper does not?

Please show me the solution -- I give up

No Incest Here!

A certain man declared to his friends one day that he had an Uncle and a nephew, both named "George". "Isn't that confusing?" replied one of the friends. "Not at all", replied the man. "They're the same person!"

Is this even possible without being involved in an incestuous relationship or marrying one's cousin?

Please show me the solution -- I give up

A Chess Puzzle

Here is a chessboard on which the players have reached an unlikely position. If you were playing the white pieces, you might go for an immediate mate, but that is too easy. Of course you might go about it more elegantly, moving each of your pieces exactly once. What are your moves?

Please show me the solution -- I give up

The King & the Coin-operated Weighing Machine

Once upon a time (or slightly thereafter) there lived a mean and miserly king who collected ten bars of silver in taxes from each of his ten duchies every year. Each bar weighed one pound. One year the tax collector arrived with the ten boxes of bars, as usual, each labeled with the name of the duke who sent it. This time, however, the tax collector looked worried.

"What troubles you, my good man?"

"Alas, Sire. I suspect one of thy dukes hath shortchanged thee, filching an ounce of silver from each bar."

"And who might that be?" roared the king.

"It is none other than Duke . . . " The tax collector fell dead, a knife thrown by an unseen hand quivering in his back. (The assassin was never found, obviously, otherwise this puzzle would be dead in the water.)

Although the king had no regular balance at his disposal, he had a coin-operated weighing machine, the ancient kind that told both your weight and your fortune when you put a penny in it. Unfortunately, he had only one penny left. How did he determine the guilty Duke?

Please show me the solution -- I give up

The Overpass

A truck driver taking a back road to avoid excess traffic came to a concrete overpass that looked a bit too low for him to pass under. Easing the mighty tractor gingerly forward, he was dismayed to hear the box scrunching on the underside of the overpass. Getting out of the cab, he inspected the roof and swore a mighty oath that went something like this:

*!%#*@#%#@!

He only needed half an inch. So near and yet so far! Nevertheless, he was soon on his way again, thanks to a very bright idea. How did he get under without further damage to his rig?

Please show me the solution -- I give up

Four Love Bugs

When a Love Bug sees another Love Bug, he/she crawls rapidly toward it to discover whether it is the opposite sex. Contact is always followed by mating or wandering off in search of a more suitable partner (the aim being to produce progeny, rather than a homosexual encounter.)

Thanks to one of those miracles that are very rare in nature (but helpful in the framing of mathematical puzzles) each of four Love Bugs finds themselves at the corners of a square that is ten centimetres on a side. Each Love Bug spots a Love Bug directly ahead of it and all four simultaneously begin to crawl towards their intended at the rate of one centimetre per second. Thus the four bugs follow a spiral trajectory, as shown.

How long will it take for all four bugs to make contact? [Warning: you can solve this problem using advanced calculus or you can just think about it for a minute.]

Please show me the solution -- I give up

How to Turn Water into Wine

Recipe: Start with one litre of wine and one litre of water, each in a two-litre jug. Now pour half the wine into the water, mix well, then pour half a litre of the wine/water mixture back into the wine. Since the half-litre poured into the second container was pure wine and since the half-litre poured back was a mixture, there will be more wine in the second container than there is water in the first. Since both jugs contain the same amount of fluid after the operation, the total amount of wine has increased, while the total amount of water has decreased.

Clearly, the operation can be expanded to produce as much wine as needed. On the other hand, something for nothing should always make one suspicious. Is there a hitch?

Please show me the solution -- I give up

Perpetumm Mobile

The dream of perpetual motion (producing a mechanism that produces more energy than it consumes) seems to be as old as humanity. Unfortunately, he law of conservation of energy rules against the possibility.

The famous English science fiction author, H. G. Wells, used a marvelous material called "cavorite" to make a voyage from Earth to the Moon. If a square metre sheet of this marvelous material is placed on the ground, it blocks all gravity from passing through it, so to speak. If one steps onto the sheet, one instantly becomes weightless. Wells used a hollow sphere to house the space voyagers. The sphere was equipped with two cavorite blinds. By drawing the earthside blind and opening the moonside one, the astronauts created a positive gravitational attraction of the sphere for the moon, and off it went, "falling" upward.

Show that cavorite must be a physical impossibility by constructing a simple perpetual motion machine that uses the same sheet of cavorite.

Please show me the solution -- I give up