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 answer is Chirag.

To solve this arrangement puzzle, we can assign positions from 1 to 7 in a clockwise direction. Since all seven children are facing the center (inside) of the circle, moving to the "left" corresponds to a clockwise direction (adding positions), and moving to the "right" corresponds to a counter-clockwise direction (subtracting positions).

Let us denote the rules for passing mathematically:

$\text{Immediately to the left}=+1\text{position}$

$\text{Immediately to the right}=-1\text{position}$

$\text{Second to the left}=+2\text{positions}$

$\text{Second to the right}=-2\text{positions}$

Let's place Bina, who starts the game, at position 1. Now we can track the positions sequentially based on the given table:

Round 1: Bina passes the buck "Immediately to the left" to Aarav.

$\text{Position of Aarav}=1+1=2$

So, Aarav sits at position 2.

Round 2: Aarav (at position 2) passes it "Second to the right".

$\text{Next receiver's position}=2-2=0$

In a 7-person circle, position 0 is equivalent to position 7. So, the person at position 7 receives the buck.

Round 3: The person at position 7 passes it "Immediately to the right" to Diya.

$\text{Position of Diya}=7-1=6$

Thus, Diya sits at position 6.

Round 6: Aarav (who is at position 2) passes it "Second to the left".

$\text{Next receiver's position}=2+2=4$

The person at position 4 receives the buck.

Round 7: The person at position 4 passes it "Immediately to the left" to Gaurav.

$\text{Position of Gaurav}=4+1=5$

So, Gaurav sits at position 5.

Round 8: Gaurav (at position 5) passes it "Immediately to the left".

$\text{Next receiver's position}=5+1=6$

Since Diya is at position 6, Diya receives the buck at the end of Round 8.

Round 9: Diya (at position 6) passes it to Farhan. The valid moves from Diya's position (6) are:

- Immediate left: Position 7

- Immediate right: Position 5 (Occupied by Gaurav)

- Second left: Position 1 (Occupied by Bina)

- Second right: Position 4

Since positions 1 and 5 are already occupied, Farhan must be at either position 4 or position 7.

Round 10: Farhan passes it to Chirag.

If Farhan were at position 7, the valid pass destinations would be positions 1, 6, 2, or 5. However, these positions are already occupied by Bina, Diya, Aarav, and Gaurav respectively, leaving no valid seat for Chirag. Therefore, Farhan cannot be at position 7 and must be seated at position 4.

Now, assuming Farhan is at position 4, his valid pass destinations are positions 5, 3, 6, and 2. Since Gaurav, Diya, and Aarav occupy positions 5, 6, and 2 respectively, the only available destination is position 3. Therefore, Chirag must be seated at position 3.

With positions 1, 2, 3, 4, 5, and 6 filled by Bina, Aarav, Chirag, Farhan, Gaurav, and Diya respectively, the final remaining position, which is position 7, must be occupied by Eshan.

The question asks: Who is sitting third to the left of Eshan?

Eshan is seated at position 7. Moving "third to the left" corresponds to adding 3 positions clockwise.

$7+3=10$

In a 7-person circle, position 10 is calculated modulo 7:

$10-7=3$

The person sitting at position 3 is Chirag, matching our earlier derivation.