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Some of the problems given below can be solved by pure logic, while others rely on the application of basic principles in physics. It is the reasoning that is more important than the answer and some of the problems are more challenging than others. I intend producing solutions at a latter date.

Logic

You are given 12 identical looking brass balls, one of which is different in mass to the others and are also provided with a beam balance. You can place any number of balls on each side of the scales, to see if they balance. How can you find the aberrant ball with with just 3 such measurements on the balance ? [You are not told if the defective ball is lighter or heavier than the others].

You are given many bags each containing identical gold coins. One of the bags however contains only fake coins, each of which is one gram heavier than the standard coin of known weight. If each bag contains differing numbers of coins, how could you determine the bag of fake coins with just one weighing?

Two centurions each guard a door, one leads to Heaven and the other leads to Hell. One of the guards always lies, while the other only speaks the truth. How can you determine which is the door to Heaven, by asking just one question, to one of the guards?

You are asked to remove 4 litres from an open barrel full of water but you are only given two containers, one of which can hold 5 litres while the other can hold 3litres. how can this be done?

An expensive game involves two players each placing a pound coin on a circular table. The last person to be able to place a coin wins the table full of money. If you are allowed to go first, how can you guarantee to win every time? ( clue --symmetry)

Is the set of all extraordinary sets, itself ordinary or extraordinary? This is Russell's paradox which can be reframed in terms of a country in which each library has an index of all its books. Some libraries also include the actual index book itself (i.e.the index book is an ordinary set), while in other libraries, the index book does not mention itself (an extraordinary set). Now the national museum contains a master index book, which lists all the index books of those libraries that do not contain themselves (i.e.a list of all the extraordinary sets). The question therefore arises as to whether this master index book should contain its own title. If it does not, then it cannot be said to contain all those index books that do not contain themselves, whereas if it does contain itself it cannot be an index only of books that do not contain themselves?

The world's most beautiful card trick? The magician asks a member of the audience to shuffle a new pack of cards and then to select 5 cards, which are given to the magician's assistant. She then reveals 4 of these cards and asks the magician to decide upon the identity of the fifth, hidden card! [The method is foolproof and relies on the mathematics of information theory]

You are presented with 3 envelopes one of which contains a cheque for a Łmillion. Having selected one of the envelopes the other person (who knows the identity of the winning envelope) discards one of his remaining 2 envelopes (one which he knows is empty) and offers to exchange envelopes with you. Knowing that the cheque is now definitely contained in one of these 2 envelopes, should you keep your original choice or exchange envelopes? [There is a big difference in the odds depending on your action]

The word “long” is heterological (since the word is actually short in length) whereas the word “short” is not heterological (since the word “short” is short). Is the word “heterological” itself heterological? [The word ACRONYM is itself an acronym but what does it stand for?]

In Mexico a US dollar is only worth 90 Mexican cents. Likewise in the USA a Mexican dollar is only worth 90 US cents. An American walks across the bridge over the Rio Grande into Mexico and buys a box of matches for 10 cents and pays with a Mexican dollar. The shopkeeper gives him a US dollar (worth 90 cents in Mexico) as change. The man then returns across the bridge to the USA and buys another box of matches out of the US dollar and receives a Mexican dollar (worth 90 US cents) as his change. He is able to repeat this process ad infinitum accumulating boxes of matches but still remaining with his original dollar. How is this possible?

A man looks into the mirror and sees the name of a well-known oil company written in red.What time is it?

Three people pay for their meal, which they are told amounts to a total of Ł30. They each give Ł10 but when the waiter goes to the till, he is told that he has overcharged and that the actual bill is only Ł25. Returning to the table he decides to inform the guests that the meal actually came to Ł27 and he returns them each Ł1 and keeps the remaining Ł2 as a tip. So the 3 guests paid Ł9 each which together with the Ł2 tip makes Ł29 in total. What happened to the remaining pound?

A man is told that the sum of 2 unknown numbers (non of which are unity), while a woman is told the product of the 2 numbers. The man says to the woman “you do not know what the 2 numbers are” to which the woman replies “I didn’t, but I do now”. The man then says “in that case so do I now know what they are”. What were the 2 numbers?

Which is the largest natural number, a Centillion or a Googelplex?

Three people in a room have a spot placed on each of their foreheads. Two of them are white spots, while one of them is black but they are unable to know which is on their own forehead, only what is on the others. They are then all told that at least one of them has a white spot on their forehead and after a brief pause, the 2 with the white spots are able to identify themselves. How is this possible?

There is a technostock market in which many (but not necessarily all) of the companies are bankrupt but none of these companies know that they themselves are insolvent, but they are aware of all those others that are. Ignorance is bliss and the stock market continues unbothered until the head of the business institute makes an announcement that at least one of the companies is bankrupt. Nothing happens at first but after a short time (depending on the size of the bankrupt market) there is frantic panic selling and the technostock market crashes. How could this happen.[The answer to this and the following problem involves an extrapolation of the problem of the 3 spots above]

In a small village in a distant land, the law states that if a woman find out that her husband has been unfaithful, she is justified in killing him. It transpires that every married man in that village is unfaithful but non of the wives are aware that their own husband's infidelity, only that everyone else's husband is unfaithful. Everything is peaceful until one afternoon, the local spiritual leader declares that at least one man in the village is unfaithful. That evening all the women in the village murder their husband. Why did this occur when the leader only said what each individual wife already knew? (see above comment)

A man is condemned to death on Saturday, and the judge decrees that the execution will take place before next Friday but that he will not know exactly which day, until midday the day before the actual hanging. The prisoner then informs the judge that he will therefore be unable to proceed with the execution! He reasons that it cannot be next Thursday, since it is the last possible day and he would therefore know his fate, before midday on the prior day. Likewise it cannot be on Wednesday, since Thursday has just been precluded and this would hence also mean that he would know the day of execution before the previous midday. This reasoning can then be continued to eliminate Tuesday , then Monday, followed by Sunday, which only leaves today Saturday, which is also not acceptable ,since he was not informed yesterday? So does the prisoner escape execution? (see underlined coment below).

Prove that the square root of a prime number, cannot be expressed as a fraction (ie. it is an Irrational number). Bertrand Russell believed that Pythagoras was intellectually "one of the most important men that ever lived " However when on of Pythagoras' students proved that he square root of 2 was irrational, this initially threatened a great deal of the Pythagorean beliefs, since whole numbers were given an almost mystical importance and it was disconcerting to realise that a quantity could not be expressed as a ratio of two whole numbers. [This can be proved by contradiction using elementary arithmetic]

Wittgenstein said that a sentence cannot refer to itself; all sentence can do is say what it means .The following are examples of self referential sentences, which therefore lead to paradoxes. Similar 'strange loops' can be found in Godels incompleteness theorem, Richard's paradox, Cantor's Paradox, Bach's fugues, Escher's drawings and in Turin's universal computer (halting problem) and is present in Russell's criticism of Frege's set theory (the set of all sets that do not contain themselves)

This sentence is false

Is this a question? {Yes, if this is an answer?}

If a man says that he is a liar should you believe him?

The sentence below is true.

The sentence above is false.

"The least integer not describable using less than 19 syllables" has only 18 syllables.

"I cannot imagine the world existing without me" {A statement which illustrates the impossibility of perceiving personal non existence}

You can generate a strange loop using this web site, by carrying out the following steps.
Return to home page and click on the "Comments" button (or Saturn image). Scroll down on the large (blue) right hand frame until you come to the "Return to home page" hyperlink and click on this link to return this frame to home page. Continue to repeat this process by clicking on the "Comments" link, scrolling down and then clicking on the "Return to home page", thus producing a series of aeroplane banners nested within each other. In other words a set of sets, each of which contains itself !!!

Physics

A man in a lift is holding a float under water in a bucket. The lift cable snaps and during the descent the man takes his hand of the float. Does the float return to the surface as normal, stay where it is or sink to the bottom of the bucket? (clue – principle of equivalence)

An “S” shaped garden sprinkler system (which usually rotates in an anticlockwise direction) is placed in a large container of water and the direction of the pump is reversed so that it now sucks in water. Does the sprinkler still rotate anticlockwise, does it move in the opposite (clockwise) direction or does it remain stationary? -Explain

A man sits on his bicycle. Does the pressure in the tyres go up, down or stay the same?

Another bicycle has its pedals placed vertically above one another and the top stirrup is tied to an inelastic wire that is attached to a wall. If you were to pedal forwards, in which direction would you move?

A glass contains ice floating in water at an equilibrium temperature of 0 degrees Centigrade. What happens to the level of the water when all the ice has melted and the glass contains only water at 0 degrees centigrade?

A block of tungsten is slowly cooled from 3000C down to minus200c. what happens to its weight.?. Explain

A car moving at constant velocity has a helium balloon tied to its handbrake. If the car suddenly brakes which direction does the balloon move, forwards, backwards or does it remain upright? (clue – principle of equivalence)

What is the colour of an Eskimos shadow?

If the Earth moves around the sun due to its large gravitational force, why is it that it is the weaker pull of the moon that is mainly responsible for the tides?

An astronaut wants to rendezvous with a space station that he sees directly ahead of him travelling at the same speed but separated by a fixed distance of several hundreds of yards. He fires his booster rocket behind him in the hope of increasing his speed but finds that this results in his rocket being further separated from the space station. Why does this happen and how should he go about closing the gap with the rocket. [This actually occurred during the first attempted space docking!]

At what angle does 2 colliding billiard balls move apart?

A cat falls upside-down from its tree, without any angular momentum, yet it manages (as always) to land on its feet, how can this happen. [This puzzled people for years before the advent of slow motion filming as the cat appeared to violate the conservation of angular momentum).

What happens when an irresistible force meets an immovable object? (the answer can be found elsewhere in another web page)

How are mystics able to walk across red-hot coals?

What rains down on you by day but passes up through your bed at night?

Why is the sky blue, the setting sun red and leaves green?

A room containing a single light bulb, has one door but no windows. Given the option of 3 switches outside the room how could you determine which one activates the light bulb, if you are only allowed to enter the room once? (You can operate the switches on and off, as many times as you want).

Is it possible to float a 100 000 ton ship in 100 pounds of water?

Two identical balls are projected horizontally along frictionless tracks to which they are secured. One of the tracks then loops upwards in a semicircle, while the other loops symmetrically downwards in a semicircle, they then continue horizontally to the finish line. If they both leave at the same time and with the same initial velocity, which ball arrives first or do they both arrive together?

On his 70^th birthday, Einstein was presented with a version of a toy cup and cone, only with a heavy brass ball connected to the cone by a spring, which is not strong enough to pull the ball into the cone. The whole cup and cone is protected by a (transparent) plastic globe and attached to a long broom handle. How can you guarantee that the cup can be caught in the cone on every attempt ? (clue – principle of equivalence)

In deep space a bullet is fired between 2 fixed points. The experiment is repeated but this time the bullet has to pass through the centre of a large planet; does the journey take more, less or the same amount of time? If the whole experiment is repeated with a beam of light, the answer is different – Explain!

You are given 2 cylinders of exactly the same dimensions, mass, and outward surface appearance. They are however made of different non magnetic materials, so that one is solid while the other is hollow. How can you simply distinguish the cylinder that is hollow?

Dirac'scissors problem; Fasten one end of a ribbon (or tie /belt) to the handle of a scissors and the other end to a solid object such as a wall. Rotate the scissors 2 revolutions about its long axis, so that the ribbon is now twisted by 720 degrees. Without severing any of the connections, how can the twist in the ribbon be undone, without rotating the scissors about any of its axes (and maintaining the direction it is pointing throughout). This method does not work when the scissors is rotated through only 360 degrees and demonstrates the fact that the group space of rotations in Euclidean space is doubly connected. For Spinors such as electrons, the group space is simply connected and this is why rotating an electron by 360 degrees does not return it to its original state but turns it upside-down!! As another illustration, take the ends of the ribbon in each hand and swap positions of your hand. Notice how the ribbon has a twist in it. This exchange of hands is equivalent to rotating each hand by 180 degrees i.e. the whole system has been transformed by 360 degrees but it has not returned to its original state (only a Euclidean rotation of 720 degrees is topologically equivalent to no rotation). The group space of spinors is however simply connected and 360 degrees does return electrons back to their original state in spinor space, so this must correspond to the 720 in ordinary Euclidean space. Hence if you exchange the position of 2 electrons (as with exchanging your hands while holding the belt), this correspond to 360 rotation in Euclidean space, which is equivalent to only 180 in the spinor space of the 2 electrons wavefunction. The wavefunction therefore changes sign i.e. the electron wavefunction is antisymmetric, meaning that 2 electrons cannot occupy the same quantum state (Pauli's exclusion principle) This is also the reason why, when an electron is rotated by one revolution, it ends up spinning upside-down. [It requires a further 360 degree rotation to return it to its original state -- this being completely at odds with the everyday experience of classical spinning objects].the statistics of an assembly of such spinors can be illustrated as described below

You are given 3 balls and are to place them on 4 steps. How many distinguishable ways are there of doing this if;

(a) Each of the balls are distinguishable (e.g. each has a different colour)

(b) Each of the balls are indistinguishable

The steps can be regarded as representing different energy levels, then the first case (a) is an example of an assembly in classical physics and leads to Boltzmann's statistics in thermodynamics. The options (b) and (c) lead to quantum statistics, the first producing bosons such as in a collection of photons in black body radiation, while the other is fermi statistics, such as the orbitals of electrons in an atom (cf. Pauli's Exclusion Principle and also spinors above).

The following 3 paradoxes require some knowledge of relativity, namely that all inertial motion (i.e. no acceleration) is relative and that when bodies are in such motion, their length contracts, time slows down and their mass increases, although these effects only become noticeable/significant at speeds close to that of light. The central tenet of Special relativity is that the laws of physics (including the constancy of the speed of light) are the same in all inertial frames. The challenge is to explain how the 3 scenarios below do not actually lead to paradoxes but only appear to do so when Relativity theory is not applied correctly.

In the year 3001, a high velocity rocket is developed on the moon, which is tested at a speed of 0.7 the speed of light, relative to engineers on the moonbase. At the same time and in the same direction as the rocket's motion, an Alien is travelling towards the moon at half the speed of light. If the Alien is approaching the moon at 0.5c and the rocket is moving across the moon at an additional 0.7c, does it mean that the rocket must be travelling at 1.2c relative to the Alien? If not why not?

Castor leaves his twin brother Pollux behind on Earth and travels in his rocket at 0.96 the speed of light. He then rapidly reverses direction and returns to Earth in the same manor. Who is older at their reunion. Since all motion is relative, how can one twin age less than the other?

A man tries to fit a 20 foot pole into a12 foot barn by running at 0.8 times the speed of light. To his friend standing next to the barn, the ladder has contracted to 12 feet and he is therefore able close the barn door when the ladder is inside the barn. However to the man running, the pole is still 20 feet long, while the barn has contracted to only 9 feet. How can they both be correct in their experiences?