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 third to the left of Eshan?

The correct option is Chirag.

To determine who is sitting third to the left of Eshan, we can model the seating arrangement of the seven children in a circle. Let the positions of the seven children in the circular arrangement be numbered 0 to 6 in a counter-clockwise direction (which represents the "left" direction, while clockwise represents the "right" direction). Therefore, the possible pass movements from a position x are defined as follows:

• Immediately to the left (IL): $x+1mod7$
• Immediately to the right (IR): $x-1mod7$
• Second to the left (2L): $x+2mod7$
• Second to the right (2R): $x-2mod7$

Step 1: Analyzing the Passes to Determine Seating Positions

We are given that the game starts with Bina passing the Buck in Round 1. Let us assign Bina's position to be 0:

$\text{Position of Bina}=0$

Round 1: Bina passes "Immediately to the left" (IL) to Aarav. Thus:

$\text{Position of Aarav}=(0+1)mod7=1$

Round 2: Aarav passes "Second to the right" (2R) to a child, let's call them X₂. Thus:

${\text{Position of X}}_2=(1-2)mod7=5$

Round 3: X₂ passes "Immediately to the right" (IR) to Diya. Thus:

$\text{Position of Diya}=(5-1)mod7=4$

Wait, if Diya is at position 4, then let us trace the later rounds to ensure consistency with Gaurav's position.

Round 6: Aarav (who receives the Buck in Round 5 at position 1) passes "Second to the left" (2L) to a child X₆. Thus:

${\text{Position of X}}_6=(1+2)mod7=3$

Round 7: X₆ (at position 3) passes "Immediately to the left" (IL) to Gaurav. Thus:

$\text{Position of Gaurav}=(3+1)mod7=4$

Wait, if Gaurav is at 4, then Diya cannot also be at position 4. Let us re-examine the modulo arithmetic or the direction definition.

Let's define "Immediately to the left" (IL) as $x-1$ and "Immediately to the right" (IR) as $x+1$ (or vice versa):

If we define Left as +1 and Right as -1:
Bina is at 0.
Round 1: Bina (0) passes IL (+1) to Aarav (1).
Round 2: Aarav (1) passes 2R (-2) to X₂ = 6.
Round 3: X₂ (6) passes IR (-1) to Diya = 5.
Let's check Round 6, 7, 8:
Round 6: Aarav (1) passes 2L (+2) to X₆ = 3.
Round 7: X₆ (3) passes IL (+1) to Gaurav = 4.
Round 8: Gaurav (4) passes IL (+1) to X₈ = 5.
Round 9: X₈ (5) passes to Farhan.
Round 10: Farhan passes to Chirag.

This layout is perfectly consistent and resolves the position of all seven children:
• Position 0: Bina
• Position 1: Aarav
• Position 2: Chirag
• Position 3: Farhan
• Position 4: Gaurav
• Position 5: Diya
• Position 6: Eshan (since Eshan is the only child left to be assigned, and position 6 was X₂)

Step 2: Finding who is third to the left of Eshan

Eshan is seated at Position 6. Since "Left" corresponds to a step of +1 in our coordinate system, we count three positions to the left (counter-clockwise) from Eshan (Position 6):

• First to the left of Eshan: $(6+1)mod7=0$ (Bina)
• Second to the left of Eshan: $(6+2)mod7=1$ (Aarav)
• Third to the left of Eshan: $(6+3)mod7=2$ (Chirag)

Thus, Chirag is sitting third to the left of Eshan.