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

For which of the following children is it possible to determine how many times they received the Buck?

The correct answer is Gaurav.

To solve this, let us map out the seating arrangement. We can number the seven seats around the circular table from 0 to 6 in a clockwise direction. Since the children are facing the inside of the circle, moving to the "left" corresponds to moving clockwise (adding to the position), and moving to the "right" corresponds to moving counter-clockwise (subtracting from the position). If a position number exceeds 6 or drops below 0, we simply wrap around the circle (modulo 7).

Let us define the position changes for each pass type:

• Immediately to the left: $+1$

• Immediately to the right: $-1$

• Second to the left: $+2$

• Second to the right: $-2$

Let the game start with Bina at position 0.

Round 1: Bina passes the buck "Immediately to the left" to Aarav. We can determine Aarav's position:

$\mathrm{Aarav}=0+1=1$

Round 2: Aarav (at position 1) passes the buck "Second to the right".

$\mathrm{Receiver\; 2}=1-2=-1$

In a 7-seat circle, position $-1$ is equivalent to position 6.

Round 3: The child at position 6 passes "Immediately to the right" to Diya.

$\mathrm{Diya}=6-1=5$

Round 6: We skip Rounds 4 and 5 for a moment because of missing pass types, but the table shows that Aarav (at position 1) receives the buck in Round 5. This means Aarav is the one passing in Round 6. He passes "Second to the left".

$\mathrm{Receiver\; 6}=1+2=3$

Round 7: The child at position 3 passes "Immediately to the left" to Gaurav.

$\mathrm{Gaurav}=3+1=4$

Round 8: Gaurav (at position 4) passes "Immediately to the left".

$\mathrm{Receiver\; 8}=4+1=5$

Since we know position 5 is occupied by Diya, Diya is the receiver in Round 8.

Let's summarize the confirmed seating positions:

• Bina is at position 0

• Aarav is at position 1

• Gaurav is at position 4

• Diya is at position 5

The unoccupied seats are 2, 3, and 6. Therefore, the remaining children (Chirag, Eshan, and Farhan) must sit in these three seats in an unknown order.

Now, let's list who received the buck in each of the 10 rounds:

• Round 1: Aarav

• Round 2: Someone at position 6 (Chirag, Eshan, or Farhan)

• Round 3: Diya

• Round 4: Unknown

• Round 5: Aarav

• Round 6: Someone at position 3 (Chirag, Eshan, or Farhan)

• Round 7: Gaurav

• Round 8: Diya

• Round 9: Farhan

• Round 10: Chirag

Gaurav receives the buck in Round 7. To see if Gaurav could have received it in Round 4 as well, we must test if he could be the Round 4 receiver. If Gaurav (at position 4) received the buck in Round 4, he would be the one passing it to Aarav (at position 1) in Round 5. Let's calculate the possible positions Gaurav can pass to from position 4:

$4+1=5$
$4-1=3$
$4+2=6$
$4-2=2$

Aarav is at position 1, which is not among Gaurav's reachable positions. Therefore, Gaurav cannot pass the buck to Aarav, making it impossible for Gaurav to be the receiver in Round 4. This confirms that Gaurav receives the buck exactly once throughout the entire game.

By contrast, we cannot be certain about the other children. For example, Bina (at position 0) could have been the Round 4 receiver because Diya (at position 5) can pass "second to the left" to reach position 0, and Bina can then pass "immediately to the left" to Aarav at position 1. Farhan and Eshan's positions are ambiguous among seats 2, 3, and 6, meaning they might have received the buck multiple times depending on where they sat.

Thus, Gaurav is the only child for whom we can definitively determine the number of times they received the buck.