Instructions [25 - 28 ]

Seven children, Aarav, Bina, Chirag, Diya, Eshan, Farhan, and Gaurav, are sitting in a circle facing inside (not necessarily in the same order) and playing a game of 'Passing the Buck'.

The game is played over 10 rounds. In each round, the child holding the Buck must pass it directly to a child sitting in one of the following positions:

• Immediately to the left;

• Immediate to the right;

• Second to the left; or

• Second to the right.

The game starts with Bina passing the Buck and ends with Chirag receiving the Buck. The table below provides some information about the pass types and the child receiving the Buck. Some information is missing and labelled as '?'.

Round	Pass Type	Received by
1	Immediately to the left	Aarav
2	Second to the right	?
3	Immediately to the right	Diya
4	?	?
5	?	Aarav
6	Second to the left	?
7	Immediately to the left	Gaurav
8	Immediately to the left	?
9	?	Farhan
10	?	Chirag

Who is sitting immediately to the right of Bina?

The correct option is Eshan.

Step-by-step Explanation:

1. Understanding the Circular Arrangement and Directions
There are 7 children sitting in a circle facing inside. Let us number the 7 positions along the circle from 1 to 7.
From the provided image, we can identify the position numbers and two known children:
• Position 1 is occupied by Bina.
• Position 2 is occupied by Aarav.
• Position 6 is occupied by Diya.
• Positions 3, 4, 5, and 7 are initially unknown.

Since the children are facing inside the circle:
• Moving left (clockwise) from a position corresponds to increasing position numbers.
• Moving right (counter-clockwise) corresponds to decreasing position numbers.

Thus, the pass types correspond to the following transitions (modulo 7):
• Immediately to the left:
$$+1$$
• Immediately to the right:
$$-1$$
• Second to the left:
$$+2$$
• Second to the right:
$$-2$$

2. Tracing the Passes in the Table
• Start: The game starts with Bina at Position 1.
• Round 1: Bina (1) passes "Immediately to the left" (+1) to Aarav at Position 2 (since 1 + 1 = 2). This matches the table.
• Round 2: Aarav (2) passes "Second to the right" (-2). The recipient is at position:
$$2-2=7$$
So, the child at Position 7 receives the Buck.
• Round 3: The child at Position 7 passes "Immediately to the right" (-1). The recipient is at position:
$$7-1=6$$
This matches the table, which states the Buck is received by Diya (Position 6).

3. Analyzing Rounds 6 to 8 to Place Gaurav
• Round 6: The pass is "Second to the left" (+2) to an unknown child.
• Round 7: The unknown child passes "Immediately to the left" (+1) to Gaurav.
• From Round 5, we know the Buck is with Aarav (Position 2).
• In Round 6, Aarav passes it from Position 2 using "Second to the left" (+2), so the Buck goes to:
$$2+2=4$$
• In Round 7, the child at Position 4 passes it using "Immediately to the left" (+1), so the Buck goes to:
$$4+1=5$$
According to the table, the child who receives the Buck in Round 7 is Gaurav. Therefore, Gaurav is sitting at Position 5.

4. Determining the Positions of the Remaining Children
So far, we have established the following positions:
• Position 1: Bina
• Position 2: Aarav
• Position 5: Gaurav
• Position 6: Diya

The remaining unoccupied positions are 3, 4, and 7.
The remaining children to place are Chirag, Eshan, and Farhan.

Let us trace the final rounds:
• In Round 8, the child at Position 5 (Gaurav) passes "Immediately to the left" (+1), so the Buck goes to Position 6 (Diya).
• In Round 9, Diya (Position 6) passes the Buck to Farhan (who must be at one of the remaining positions: 3, 4, or 7).
• In Round 10, Farhan passes the Buck to Chirag (who must also be at one of the remaining positions).

Let's check the possible moves from Diya's position (6) to a valid position in {3, 4, 7}:
• If Diya passes to Position 4 (a shift of -2), Farhan is at Position 4.
• If Diya passes to Position 7 (a shift of +1), Farhan is at Position 7.

Now we analyze where Farhan can pass in Round 10 to reach Chirag (who must also be at one of the remaining positions):
• Case 1: Farhan is at Position 7.
From Position 7, the possible passes are to:
• Position 6 (Diya)
• Position 5 (Gaurav)
• Position 1 (Bina)
• Position 2 (Aarav)
None of these positions are in the remaining set {3, 4}. Therefore, Chirag cannot receive the Buck if Farhan is at Position 7.

• Case 2: Farhan is at Position 4.
From Position 4, the possible passes are to:
• Position 3 (shift of -1)
• Position 2 (shift of -2)
• Position 5 (shift of +1)
• Position 6 (shift of +2)
Position 3 belongs to the remaining set {3, 7}. Thus, Chirag must be at Position 3.

With Farhan at Position 4 and Chirag at Position 3, the final remaining position (Position 7) must be occupied by Eshan.

5. Conclusion
The final seating arrangement is:
• Position 1: Bina
• Position 2: Aarav
• Position 3: Chirag
• Position 4: Farhan
• Position 5: Gaurav
• Position 6: Diya
• Position 7: Eshan

Looking at the circle, the position immediately to the right (counter-clockwise) of Bina (Position 1) is Position 7, which is occupied by Eshan.