A Round Table Conference

8 people have to be seated at a round banquet table. The seats are number 1-8. 

For this teaser, 5 is opposite 1, 6 is opposite 2 and so on. Likewise, "next to" means one of the neighboring seats only. i.e., 8 is next to 1 and 7.
A and H are the only French speaking people on the table. They need to be seated together. 

B and F should sit opposite each other. 

C, F, and G all know Russian, but don't necessarily need to sit next to each other. In fact, G wants to sit next to someone who knows French. 

D is English and insists on sitting next to at least one other English speaking person, and pat opposite the good looking H. E is the only other English speaker in the group, but he wants to sit next to someone who knows Russian. 

G agrees to sit next to F only on condition that the other side must have a French speaker. C does not sit next to either of them. 

C however, agrees to sit next to B, who is the only Bavarian in the group.
A will not sit next to a Russian or an English speaker.

THIS should be the seating arrangement! 

Seating Arrangement in A Round Table Conference

What was the arrangement challenge?

Given Data : 

8 people have to be seated at a round banquet table. The seats are number 1-8.

1. A and H are the only French speaking people on the table. They need to be seated together. 

2. B and F should sit opposite each other. 

3. C, F, and G all know Russian, but don't necessarily need to sit next to each other. In fact, G wants to sit next to someone who knows French. 

4. D is English and insists on sitting next to at least one other English speaking person, and pat opposite the good looking H. 

5. E is the only other English speaker in the group, but he wants to sit next to someone who knows Russian. 

6. G agrees to sit next to F only on condition that the other side must have a French speaker. C does not sit next to either of them. 

7. C however, agrees to sit next to B, who is the only Bavarian in the group.
A will not sit next to a Russian or an English speaker.

Arranging STEPS :

1] From (1) & (3), it's clear that G should be seating next to the A or H. That is G, A and H are occupying adjacent seats whose order is yet to be known. 

2] Now as per (2), B and F must be on opposite seats, they need to 'surround' the group of G - A - H. That's the only way they would be opposite. (There are 2 ways that they can do this and this would result into 2 possible solutions as concluded in conclusion).

3]  As per (6), G is ready to sit next to F so that G and F should be occupying adjacent seats. Order of seat to be occupied by A and H are yet to be known.

4] As per (7), C can seat next to B. A can't seat next to Russian G. 

5] As per (4) and (5), D & E should be next to each other but as per (4) D has to be opposite to H. 

CONCLUSION : 

In order to make seat arrangement in accordance with everyone's preference, we just need to ask them to seat in alphabetical order of their names. Here, seat number doesn't matter and what matters is that everyone is getting their favorite seat.

Interestingly, they can seat in circle clockwise or anticlockwise according to alphabetical order of their names.

 

Four Glasses Puzzle

Four glasses are placed on the corners of a square table. Some of the glasses are upright (up) and some upside-down (down). A blindfolded person is seated next to the table and is required to re-arrange the glasses so that they are all up or all down, either arrangement being acceptable, which will be signaled by the ringing of a bell. 

The glasses may be re-arranged in turns subject to the following rules. 

1.Any two glasses may be inspected in one turn and after feeling their orientation the person may reverse the orientation of either, neither or both glasses.

2.After each turn the table is rotated through a random angle. 

3.The puzzle is to devise an algorithm which allows the blindfolded person to ensure that all glasses have the same orientation (either up or down) in a finite number of turns. The algorithm must be non-stochastic i.e. it must not depend on luck.

Here is that algorithm!

Solution of Blind Bartender's Problem

What was the puzzle?

Below is the algorithm which makes sure the bell will ring in at most five turns.

1.On the first turn choose a diagonally opposite pair of glasses and turn both glasses up.
At this point, the position of other 2 glasses is not known.

2.On the second turn, choose 2 adjacent glasses. One of them was turned up in the previous step, so other may or may not in up position. If the other is down then turn it up and if remaining one X is also in up position then bell will be rung.

If the bell does not ring then there are now three glasses up and one down(3U and 1D).

3.On the third turn choose a diagonally opposite pair of glasses. If one is down, turn it up and the bell will ring.

And if you find both are up, then you must have chosen other diagonally opposite pair.

If so, then turn one down so that 2 glasses are up and other 2 are down.

4.On the fourth turn choose two adjacent glasses and reverse both. If both were in the same orientation then the bell will ring. 

And in case, if you find one is up and other down like -

still reverse orientation of both as - 

Now diagonally opposite pairs are either up or down.

5.On the fifth turn choose a diagonally opposite pair of glasses and reverse both.

The bell will ring for sure.

   

Case of 3 Identical Notebooks

Three people all set down their identical notebooks on a table. On the way out, they each randomly pick up one of the notebooks. What is the probability that none of the three people pick up the notebook that they started with?

That's correct probability!

Probability in Case of 3 Identical Notebooks

What was the case?

Let's name peoples as Person - 1, Person - 2, Person - 3 and their notebooks as Notebook - 1, Notebook - 2 and Notebook 3 respectively.

Now there can be 6 ways 3 notebooks can be distributed among 3 persons like below.

(Here, for convenience, 3 different colors are assigned to the notebooks of 3 persons.)

As we can see, there are only 2 cases, where the each of person not getting own notebook. In rest of cases, at least 1 person got own notebook & notebooks of others are shuffled between 2.

So the probability that none of the three people pick up the notebook that they started with is 2/6 = 1/3. 

Time Taken For The Journey

A RED ant is sitting on one side of a table (point X) and a BLACK ant is sitting on the opposite side of the table (point Y).

Now both of them decides to exchange their places and starts crawling. On the way, both of the ants meet and after that, it takes 20 seconds for the RED ant to move to point Y and it takes 5 seconds for the BLACK ant to reach point X.

Find out the total time taken by the RED and the BLACK ant to make the journey.

Here are steps to calculate! 

Source 

Calculation of Time For The Journey

What was the journey?

Let the speed of RED ant is R & that of BLACK ant is B

Let time taken by them to meet be T.

Now we will apply the basic formula of distance:

Distance = Speed * Time.

The RED ant travels R T distance before meeting and 20 R after the meeting.

The BLACK ant travels B T distance before meeting and 5 B after the meeting.

Now as per the question,The distance traveled by RED ant before they both meet will be equal to the distance covered by BLACK ant after they meet. We can say the same for the vice versa case as well.

Thus,

RT = 5B and BT = 20R i.e. B = 20R/T, putting in RT = 5B

R T = [20R/T] * 5

RT = 100R/T

T^2 = 100

T = 10.

Thus the RED ant will require 10 + 20 = 30 seconds to travel the distance.

And the BLACK ant will take 10 + 5 = 15 seconds to travel the distance.