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# Arrangements

Circular Arrangements 1

__Circular Arrangements: __

Again, before we jump into solving sets, there are a certain terminologies that we need to understand first.

__1. Six persons are sitting along a circular table.__

First we draw a circle and mark six points equally spaced.

If there were eight persons sitting along a circular table, then mark eight equally spaced points.

__2. P is sitting to the right of Q __

Unlike a linear arrangement, when we say P is sitting to the right of Q, then it necessarily implies that P is to the immediate right of Q and similarly if P is to the left of Q, then it implies that P is to the immediate left of Q.

__3. P is two places to the left or right of Q__

__4____. P is sitting opposite to Q__

If it is given that P and Q are sitting opposite to each other, then it implies that they are sitting diagonally opposite to each other. i.e. there are equal number of persons between them taken in any direction i.e. clockwise or anticlockwise. Sometimes it is also said that they are farthest from each other or A and B are sitting across the table.

**Example 1:** Eight Sales executives – Aditya, Bhushan, Chintan, Deepanshu, Eshwar, Farishta, Gopal and Hemant are sitting around a circular table to discuss the monthly sales targets set by their team leader. All of them were facing the centre. The following information is known about them :

1. Hemant is two places to the left of Gopal while there are exactly two persons between Aditya and Gopal.

2. Neither Bhushan nor Aditya is the neighbor of Hemant.

3. Chintan is diagonally opposite to Deepanshu who is not the neighbor of Gopal.

**1.** If Farishta is the neighbor of Deepanshu, then who is sitting to the left of Gopal?

A. Chintan B. Eshwar C. Bhushan D. Cannot be determined

**2.** If Farishta is the neighbor of Hemant, then who is to the right of Aditya?

A. Bhushan B. Eshwar C. Deepanshu D. Cannot be determined

**3.** If Gopal and Aditya interchanged their places and Eshwar is diagonally opposite to Aditya while the other conditions remain the same, then who is two places to the left of Eshwar?

A. Bhushan B. Gopal C. Farishta D. None of these

**4.** Which of the following conditions would be sufficient to complete the entire seating arrangement of these sales executives?

A. Farishta is sitting two places to the left of Chintan.

B. Eshwar is diagonally opposite to Aditya.

C. Bhushan is three places to the left of Deepanshu.

4. Eshwar is two places to the right of Bhushan.

Solution: Let’s denote these sales executives by A, B, C, D, E, F, G and H

From statement (1) we get

Case 1 Case 2

From statement (2) as Hemant is not the neighbor of Aditya, so case 2 gets ruled out.

We get:

From statement (3), we get C and D are diagonally opposite to each other while D is not the neighbor of G.

1. Given that Farishta is the neighbor of Deepanshu and as B cannot be the neighbor of H, so the arrangement becomes:

So, Eshwar is sitting to the left of Gopal. Hence, option (B) is correct.

2. Given that Farishta is the neighbour of Hemant, So, either Eshwar or Bhushan is to the right of Aditya. Thus, option (D) is correct.

3. Given that Gopal and Aditya interchanged their places and Eshwar is diagonally opposite to Aditya,

So, Bhushan is two places to the left of E. Thus, option (A) is correct.

4. (1) The first statement that Farishta is sitting two places to the left of Chintan. The arrangement becomes:

The arrangement is incomplete as we cannot be certain of the positions of Bhushan and Eshwar.

(2) The second statement that Eshwar is diagonally opposite to Aditya would give the arrangement:

It is still incomplete as we cannot be certain of the positions of Bhushan and Farishta.

(3) The third statement that Bhushan is three places to the left of Deepanshu would give the arrangement as

.

This time we are not certain about the positions of Eshwar and Farishta.

(4) The fourth statement that Eshwar is two places to the right of Bhushan will give the arrangement as:

Thus, option (D) is correct.

1. Which gift was brought by Veronica?

a. Tie b. Pen **c. Coffee Mug ** d. Watch

2. Which gift is brought by the one who sits to the right of Cheryl?

**a. Tie** b. Pen c. Sun glasses d. Coffee Mug

3. Who brought pen?

a. Reggie **b. Jughead** c. Betty d. Veronica

4. Who is sitting to the right of Betty?

a. Jughead b. Archie c. Veronica **d. Cannot be determined**

**Solution:** From statement (2) we get :

From (3) we get Veronica is sitting two places to the right of Reggie who is sitting next to Cheryl. Now there are two cases for this:

From (1) we get that Cheryl is sitting next to the one who brought tie. In both the cases it comes out to be Reggie, and also Cheryl is sitting opposite to the one who brought coffee. Again in both these cases it is Veronica who is opposite to Cheryl and so she brought coffe mug. The arrangement further becomes:

From (4) we get Betty is sitting two places to the left of the one who brought sun glasses and herself did not bring pen.So, this implies Cheryl brought sun glasses. Also, if Betty didn’t bring pen, so she must have brought watch and so Jughead got pen for Mr Weatherbee. The final arrangement becomes:

**1. Veronica brought coffee mug. So, option (C) is correct. **

**2. Reggie is sitting to the right of Cheryl and he brought Tie. So, option (A) is correct. **

**3. Jughead brought pen. So, option (B) is correct. **

**4. Either Archie or Jughead is sitting to the right of Betty. So, option (D) is correct.**

**1.** What is the maximum number of persons that sit between Sharma and Mukherjee?

A. Two B. One C. Three D. Four E. None of these

**2.** Who sits between Kapoor and Mittal?

A. Chawla B. Sharma C. Mukherjee D. Aggarwal E. None of these

**3.** Who is the female other than Mittal and Kapoor?

A. Aggarwal B. Verma C. Chawla D. Nigam E. Cannot be determined

**4.** Who sits exactly opposite to Aggarwal?

A. Verma B. Aggarwal C. Nigam D. Kapoor E. Cannot be determined

**Solution:** From statement (3) we get between Nigam and Sharma sits Verma with Sharma to the right of Verma.

From statement (4) we get, Kapoor is a lady who is opposite to Sharma who is a male, and as also no two females are together, so we get

From statement (2), the second case gets ruled out and so we get:

From (5), we get One female who is not Mukherjee sits exactly between Sharma and Chawla. So,

**1. There are three persons sitting between Sharma and Mukherjee. So, option (C) is correct. **

**2. Mukherjee sits between Kapoor and Mittal. So, option (C) is correct.**

**3. Aggarwal is the other female. So, option (A) is correct. **

In some types of sets, the persons sitting along a circular table are not facing the centre or some may be facing the centre while others may be away from the centre. So, how to tackle such types of sets. Let’s take an example to understand.

**Example 4 :** 8 friends are sitting on a circular table facing inside and outside alternately.

Person facing the North cannot see the person facing South and vice versa. Similarly, a person facing the East cannot see the person facing the West and vice versa. No one can see himself, and rest everybody can see everybody.

The names are: P, Q, R, S, T, U, V, W and the color of their T-shirts are: Blue, Red, Black, Green, Yellow, Orange, Brown, White, not necessarily in the same order.

1. Only 2 persons whose color starts with the same letter are together, one of them is facing North.

2. U is 3 places to the right of Orange

3. Those wearing White, orange, blue, green T-shirts are facing inside.

4. W and V are opposite to each other but V gossips about W’s untidy hair with his neighbor.

5. Only one person’s name and color start with the same letter.

6. Those wearing Orange, brown and white T-shirts are sitting together, one of them facing west.

7. Red is facing south.

8. V, R, T are best friends and always sit together, one of them is wearing a blue T-shirts.

9. P is not between Q and S

10 . The ones wearing the orange and white coloured T-shirts are not adjacent

11. R is to the left of the one wearing the Red T-shirt.

**1.** Where is P facing?

A. West B. North C. North-West D. North or West

**2.** Who is 3 places to the right of T?

A. S B. W C. the person wearing White D. both B,C

**3.** Who is the neighbour of Q?

A. R B. T C. S D. U

**Solution:** As, the one facing North cannot see the one facing south and similarly the one facing west cannot see the one facing east and as they are facing inside and outside alternately, we get :

From 5 we get , one person whose name starts with the same letter as the colour of his T-shirt. So it could either be R- Red or W- White. But from (11) we get R is to the left of the one wearing the red T-shirt. So W is definitely wearing the White T-shirt.

Also if we combine (7) with (11) we, get:

From (4) we get V and W must be diagonally opposite to each other but as they can see each other, so they must be sitting at positions that are facing inwards. Combining (4) and (8) we get two cases:

From (6) we get, Those wearing Orange, brown and white T-shirts are sitting together, one of them facing west and Also from (3) we get Orange must be facing inwards. This rules out case (2), we now get:

From (2) we get, U is sitting three places to the right of The one wearing orange. So, U is wearing the red colour T-shirt. From (1) we get , only 2 persons whose color starts with the same letter are together, one of them is facing North. So, these two colours must be black and blue, but as the blue colour is facing inwards, so we get :

Also, as the one wearing the green T-shirt is facing inwards, so we get R is wearing the green T-shirt and so T is wearing the yellow T-shirt. Also as P is not between Q and S. So, this implies P is not wearing the orange T-shirt. So, final arrangement becomes:

**1. P is either facing the north or the west direction. So, option (D) is correct.**

**2. W is three places to the right of T and as he is wearing a white T-shirt, so option (D) is correct.**

**3. Definitely R, T and U cannot be the neighbours of Q, so S is be a neighbor Q. Thus, option (C) is correct. **

**Example 5 :** There is a square table which has seats around it in the following manner:

The people sitting are P, Q, R, S, T, U, V, W

1, 2, 2, 3 people are sitting in each row, not necessarily in the same order.

One can talk only to the person opposite to him and the person adjacent to him. 2 people on the same corners of different rows will not be considered adjacent. Also those who can talk will definitely talk.

There is no conversation across side A and side C.

Side D has more number of people than side B and T sits alone.

Q an S are adjacent to each other.

P, U, T do not talk to anyone.

One person sits on side C.

V is opposite Q and talks to 3 people in total.

**1.** On which side does W sit?

A. A B.B C. D D. CBD

**2.** One which side the central seat is necessarily empty?

A. A B. B C. D D. CBD

**3.** Who is opposite S?

A. R B. W C. V D. either R or W

**4.** Who sits on row A?

A. T B. P C. U D. Both B and C

**Solution:** Only one person sits on side C and we know T sits alone. So, T is seated on side C. Also as D has more number of persons seated compared to side B and 1 person is sitting on side C. So, this implies that 3 persons are sitting alongside D, 2 alongside B, two alongside A and one along side C.

Also the maximum number of persons to which a person can talk to is 3 ( two of its neighbours and one opposite) and that position is the central position along any side. For this to happen, the positions adjacent to it must be occupied. This can happen only along the side D where all the three positions are occupied. So, V is sitting in the central seat along the side D and Q is opposite to it to side B.

As S is sitting adjacent to Q and as T is sitting along side C, we get

As P, U and T do not talk to each other, so P and U cannot be on side D ( as otherwise they would have been talking to V). So definitely R and W are sitting on side D which further implies P and U are sitting along the side A. Also P and U must not be adjacent to each other otherwise they would have talked to each other. So final seating arrangement becomes:

**1. W is sitting on side D. So, option (C) is correct.**

**2. The central seat is empty on side A. So, option (A) is correct.**

**3. Either R or W is opposite S. So, option (D) is correct.**

**4. Both P and U are sitting along the side A. So, option (D) is correct. **