Logic Puzzles 5 [With Answers]

Puzzle-1:

You work at a fruit factory. 

There are 3 crates in front of you. One crate contains only apples. One crate contains only oranges. The other crate contains both apples and oranges. 

And each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples and oranges".
But the labeling machine has gone crazy and is now labeling all boxes incorrectly.
If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you label ALL of the crates correctly?

Answer: Take a piece of fruit from the "apples and oranges" crate. If it's an apple then you know that is the "apples" crate since ALL THE CRATES ARE LABELED INCORRECTLY. This means the crate marked "apples" must be "oranges" and the crate marked "oranges" must be "apples and oranges".

Puzzle-2:

A man is caught on the King's property. He is brought before the King to be punished.

The King says, "You must give me a statement. If it is true, you will killed by lions. If it is false, you will be killed by trampling of wild buffalo."

But in the end, the King has to let the man go.

What was the man's statement?

Answer: "I will be killed by trampling of wild buffalo."

If you think about it carefully: the King can't say it is true, because if so the man should be killed by the lions. But if he is killed by the lions, then his statement would be false, and so should have been trampled by buffalo.

So the King can't tell if it is a lie or truth, so decides to just let him go.

Puzzle-3:

A man works on the 10th floor and always takes the elevator down to ground level at the end of the day.

Yet every morning, he only takes the elevator to the 7th floor and then walks up the stairs to the 10th floor, even when in a hurry.

But he will go all the way to the 10th floor when others are in the elevator with him or on a rainy day.

Why?

Answer: He is too short to reach the "10" button.

But when others are with him, he can ask them to push the button for the 10th floor. 

And on a rainy day he has an umbrella, so he can push the button using that.

Puzzle-4:

There are five gears connected in a row, the first one is connected to the second one, the second one is connected to the third one, and so on.

How much faster would the last gear be if the second gear was twice the size of the first gear, and all the other gears were the same size as the first gear?

Answer: The last gear would be the same speed.

Puzzle-5:

There are three people (Alex, Brook and Cody), one of whom is a knight, one a knave, and one a spy.

The knight always tells the truth, the knave always lies, and the spy can either lie or tell the truth.
Alex says: "Cody is a knave."

Brook says: "Alex is a knight."

Cody says: "I am the spy."

Who is the knight, who the knave, and who the spy?

Answer:

Alex is a Knight

Brook is a Spy

Cody is a Knave

Brook is not the knight, since if he is, then Alex would also be the knight.

Cody is not the knight, since his statement would then be a lie. 

Therefore Alex is the knight. Hence Cody is the knave, and Brook is the spy.

Total Internal Reflection

When light travels from an optically denser medium to a rarer medium at the interface, it is partly reflected back in to the same medium and partly refracted to the second medium. This reflection is called the internal reflection.

When a ray of light enters from a denser medium to a rarer medium. It bends away from the normal, for example, the ray AO₁B in figure. The incident ray AO₁ is partially reflected (O₁C) and partially transmitted (O₁B) or reflected, the angle of refraction (r) being larger than the angle of incidence (i) As the angle of incidence increases, so does the angle of refraction, till for the ray AO₃. The angle of refraction is π/2. The refracted ray is bent so much away from the normal that it grazes the surface at the interface between the two media. This is shown by the ray AO₃D in figure. If the angle of incidence is increased still further (e.g., the ray AO₄), refraction is not possible, and the incident ray is totally reflected.

This is called total internal reflection, when light gets reflected by a surface, normally some fraction of it gets transmitted. The reflected ray, therefore, is always less intense than the incident ray, howsoever smooth the reflecting surface may be. In total internal reflection, on the other hand, no transmission of light takes place.

The angle of incidence corresponding to an angle of refraction 90⁰ say ∠AO₃N is called the critical angle (i_c) for the given pair of media. We see from Snell’s law  that if the relative refractive index is less than one then, since the maximum value of  is unity, there is an upper limit to the value of sin i for which the law can be satisfied, that is, i = i_c such that sin i_c = n₂₁.

Logic Puzzles 5

Puzzle-1:

You work at a fruit factory. 

There are 3 crates in front of you. One crate contains only apples. One crate contains only oranges. The other crate contains both apples and oranges. 

And each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples and oranges".
But the labeling machine has gone crazy and is now labeling all boxes incorrectly.
If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you label ALL of the crates correctly?

Puzzle-2:

A man is caught on the King's property. He is brought before the King to be punished.

The King says, "You must give me a statement. If it is true, you will killed by lions. If it is false, you will be killed by trampling of wild buffalo."

But in the end, the King has to let the man go.

What was the man's statement?

Puzzle-3:

A man works on the 10th floor and always takes the elevator down to ground level at the end of the day.

Yet every morning, he only takes the elevator to the 7th floor and then walks up the stairs to the 10th floor, even when in a hurry.

But he will go all the way to the 10th floor when others are in the elevator with him or on a rainy day.

Why?

Puzzle-4:

There are five gears connected in a row, the first one is connected to the second one, the second one is connected to the third one, and so on.

How much faster would the last gear be if the second gear was twice the size of the first gear, and all the other gears were the same size as the first gear?

Puzzle-5:

There are three people (Alex, Brook and Cody), one of whom is a knight, one a knave, and one a spy.

The knight always tells the truth, the knave always lies, and the spy can either lie or tell the truth.
Alex says: "Cody is a knave."

Brook says: "Alex is a knight."

Cody says: "I am the spy."

Who is the knight, who the knave, and who the spy?

Integral Function and its Properties

Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by  for all x ϵ [a, b] is called the integral function of the function f(x).

Property I: The integral function of an integrable function is always continuous.

Property II:  if φ(x) is differentiable on (a, b) and φ’(x) = f(x) for all x ϵ (a, b).

Property III: The integral function of an odd function is an even function.

If f(x) is an odd function, then  is an function.

Example: Find the greatest value of  in the interval [5π/3, 7π/4]

Solution:  We have,

⇒ F’(x) = 6 cosx - 2sinx

For all x ϵ [5π/3, 7π/4], we have

⇒ cos x > 0 and sin x < 0

⇒ F’(x) = 6 cosx - 2sinx > 0

⇒ F(x) is an increasing function on [5π/3, 7π/4]

⇒ F(x) attains greatest value at x = 7π/4.

Hence,

Greatest value = F (7π/4)

= 3√3 - 2√2 - 1.

Logic Puzzle 4 [With Answers]

Puzzle-1: You have a basket containing ten apples. You have ten friends, who each desire an apple. You give each of your friend’s one apple.

Now all your friends have one apple each, yet there is an apple remaining in the basket.
How?

Answer: You give an apple each to your first nine friends, and a basket with an apple to your tenth friend.

Each friend has an apple, and one of them has it in a basket. (Alternative answer: one friend already had an apple and put it in the basket.)

Puzzle-2: You have three bags, each containing two marbles. Bag A contains two white marbles, Bag B contains two black marbles, and Bag C contains one white marble and one black marble.

You pick a random bag and take out one marble.

It is a white marble.

What is the probability that the remaining marble from the same bag is also white?

Answer: 2/3 (not 1/2).

You know that you do not have Bag B (two black marbles) so there are three possibilities you chose Bag A, first white marble. The other marble will be white

You chose Bag A, second white marble. The other marble will be white

You chose Bag C, the white marble. The other marble will be black

So 2 out of 3 possibilities are white.

Why not 1/2? You are selecting marbles, not bags.

Puzzle-3: How many steps are required to break an m × n sized bar of chocolate into 1 × 1 pieces? 

You can break an existing piece of chocolate horizontally or vertically. 

You cannot break two or more pieces at once (so no cutting through stacks).

Answer: You need m×n - 1 steps.

By breaking an existing piece horizontally or vertically, you merely increase the total number of pieces by one. 

You already have 1 piece, so need m×n - 1 steps to get to m×n pieces.

Puzzle-4: A man is stranded on an island covered in forest.

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire)

Answer: The man picks up a piece of wood and lights it from the fire on the west end of the island.
He then quickly carries it near the east end of he island and starts a new fire. The wind will cause that fire to burn out the eastern end and he can then shelter in the burnt area.

The man survives the fire, but dies of starvation, with all the food in the forest burnt.

Puzzle-5: You are at an unmarked intersection ... one way is the City of Lies and another way is the City of Truth. 

Citizens of the City of Lies always lie.

Citizens of the City of Truth always tell the truth.

A citizen of one of those cities (you don't know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?

Answer: You ask "In which of those two directions do you live?"

A Citizen of the City of Lies will point to the City of Truth.

A Citizen of the City of Truth will point to the City of Truth.

Types of Polymerization Reactions

1. Addition polymerization (or) chain growth polymerization:

⇒ The molecules of the same monomer or different monomers add together on a large scale to form a polymer.

⇒ The monomers used are unsaturated compounds.

Eg: Alkenes, alkadienes and their derivatives.

⇒ Can take place through the formation of either free radicals or ionic species.

Free radical mechanism: A variety of alkenes or dienes and their derivatives are polymerized in the presence of a free radical generating initiator (catalyst) like benzoyl peroxide, acetyl peroxide, tert-butyl peroxide etc.,

Eg: Polymerisation of ethene to polythene.

The free radical mechanism involves following steps.

Chain initiation step:

Ċ₆H₅ - CH₂ = CH₂ → 2C₆H₅ - CH₂ - ĊH₂

Chain propagation step:

C₆H₅ - CH₂ - ĊH₂ + CH₂ = CH₂ → C₆H₅ - CH₂ - CH₂ - CH₂ - ĊH₂ → C₆H₅ (- CH₂ - CH₂-)_n – CH₂ - ĊH₂

Chain terminating step: For termination of the long chain, these free radicals can combine in different ways to form polythene.

Eg: C₆H₅ - (- CH₂ - CH₂ -)_n - CH₂ - ĊH₂ + C₆H₅ - (- CH₂ - CH₂ -)_n - CH₂  - ĊH₂ → C₆H₅ - (- CH₂ - CH₂ -)_n - CH₂ - CH₂ - CH₂ - CH₂ - (- CH₂ - CH₂ -)_n - C₆H₅

Preparation of some important addition polymers:

a) Polythene: These are of two types.

i) Low density polythene (LDP):-

⇒ By polymerization of ethene

⇒ Temperature: 350 K to 570 K in the presence of traces of dioxygen or peroxide initiator.

⇒ LDP obtained through the free radical addition and H atom abstractions has highly branched structure.

⇒ It is chemically inert and tough but flexible and a poor conductor of electricity.

ii) High density polythene (HDP):-

⇒ Addition polymerization of ethene in a hydrocarbon solvent in the presence of Ziegler - Natta Catalyst.

These are two sets of Zieglar - Natta catalyst/ co - catalyst systems. Either way, we have four chlorine atoms.

⇒ Temperature: 333 K to 343 K

⇒ Pressure of 6 - 7 atm.

⇒ HDP molecules consists linear molecules, its high density is due to close packing. It is also chemically inert and more tough and hard.

⇒ It is used for manufacturing of buckets, bottles, pipes, etc.,

b) Polytetrafluoro ethene (Teflon):

⇒ By heating tetrafluoroethene with a free radical or per sulphate catalyst at high pressure.

⇒ It is chemically inert and resistant to attack by corrosive reagents.

⇒ It is used in making oil seals and gaskets.

c) Polyacrylonitrile:

⇒ The addition polymerization of acrylonitrile in the presence of a peroxide catalyst leads to the formation of polyacryionitrile.

⇒ It is used as a substitute for wool in making commercial fibres as Orlon or acrilan.

2. Condensation polymerization or step Growth polymerization:

⇒ Involves a repetitive condensation reaction between two bifunctional monomers.

⇒ Results in the loss of simple molecules as water, alcohol, etc.

⇒ The product of each step is again a bi functional species and the sequence of condensation goes on.

⇒ So this process is also called as step growth polymerization.

Some important condensation polymerization reactions characterized by their linking units.

a. Polyamides:

⇒ Possess amide linkages.

⇒ Prepared by condensation polymerization of diamines with dicarboxylic acids and also of amino acids and their lactums.

Nylon 6, 6: Prepared by the condensation polymerization of hexamethylenediamine with adipic acid under high pressure and at high temperature.

Uses: It is used in making sheets, textile industry.

Nylon 6: Heating caprolactum with water at a high temperature.

Uses: It is used in the manufacture of tyre cords, fabrics and ropes.

b. Polyesters: Poly condensation products of dicarboxylic acids and diols.

Eg: Dacron or terylene.

It is manufactured by heating a mixture of ethylene glycol and terephthalic acid at 420 to 460 k in the presence of Zinc acetate antimony trioxide catalyst.

c. Phenol - formaldehyde polymer:

⇒ These are obtained by condensation reaction of phenol with formaldehyde in the presence of either acid or base catalyst.

⇒ The reaction starts with the initial formation of o – and/ or p – hydroxylmethylphenol derivatives, which further react with phenol to form compounds having rings joined to each other through – CH₂ groups.

⇒ The initial product could be a linear product- Novolac

⇒ Novolac is used in paints.

Bakelite: Novolac on heating with formaldehyde undergoes cross linking to form an infusible solid mass called Bakelite.

Uses: It is used in combs, phonograph records, etc.

Structure:

d. Melamine formaldehyde polymer: Melamine formaldehyde polymer is formed by the condensation polymerization of melamine and formaldehyde.

Uses: It is used in the manufacture of unbreakable crockery.

3. Copolymerization:

⇒ A mixture of more than one monomeric species is allowed to polymerize and form a copolymer.

⇒ The copolymer can be made not only by chain growth polymerization but by step growth polymerization also.

Example: Mixture of 1, 3 - butadiene and styrene can form a copolymer.

⇒ Copolymers have properties quite different from homopolymers.

⇒ For example, butaiene - styrene copolymer, is quite tough and is a good substitute of natural rubber. It is used in manufacture of auto tyres, footwear components, cable insulation etc.