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

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

Correct Answer: Gaurav

To determine which child's total number of buck receptions can be uniquely determined, we can model their seating arrangement and trace the path of the Buck through the 10 rounds of the game.

Step 1: Understanding the Seating Positions and Passes
Let the 7 children be seated at positions numbered 1 to 7 in clockwise order around the circle. Since they are facing inside the circle:
• Moving to the left corresponds to increasing the position index in the clockwise direction:

$x\to (x+1)\text{mod}7$

• Moving to the right corresponds to decreasing the position index (counter-clockwise direction):

$x���(x-1)\text{mod}7$

• Second to the left corresponds to:

$x\to (x+2)\text{mod}7$

• Second to the right corresponds to:

$x\to (x-2)\text{mod}7$

(For simplicity, we map position numbers 1 to 7 modulo 7, where 0 is treated as 7.)

Step 2: Mapping the Positions using the Table
The game starts with Bina passing the Buck. Let Bina be seated at position 1.
• Round 1: Bina (Position 1) passes Immediately to the left (which is position 2) to Aarav. Thus, Aarav is at Position 2.
• Round 2: Aarav (Position 2) passes Second to the right. This goes to position $2-2=0\equiv 7$.
• Round 3: The child at Position 7 passes Immediately to the right (which goes to position $7-1=6$) to Diya. Thus, Diya is at Position 6.
• Round 6: In Round 5, the buck is received by Aarav (Position 2). Aarav then passes Second to the left, which goes to position $2+2=4$.
• Round 7: The child at Position 4 passes Immediately to the left (which goes to position $4+1=5$) to Gaurav. Thus, Gaurav is at Position 5.
• Round 8: Gaurav (Position 5) passes Immediately to the left, which goes to position $5+1=6$ (Diya).

Step 3: Determining Possible Arrangements for the Remaining Children
The remaining three children are Chirag, Eshan, and Farhan, who must occupy positions 3, 4, and 7 in some order.
From Round 8, Diya (Position 6) passes the Buck to Farhan in Round 9. Let's analyze the valid pass types from Position 6:
1. Case I: Farhan is at Position 4.
• Diya (Position 6) to Farhan (Position 4) is Second to the right (valid).
• In Round 10, Farhan (Position 4) passes to Chirag. Chirag can be at either Position 3 (Immediately to the right) or Position 7 (Second to the left). This leaves Eshan at the remaining position (Position 7 or 3 respectively).
2. Case II: Farhan is at Position 7.
• Diya (Position 6) to Farhan (Position 7) is Immediately to the left (valid).
• In Round 10, Farhan (Position 7) passes to Chirag. For the pass to be valid, Chirag must be at Position 4 (Second to the right) or Position 1 (not possible as Bina is here). Thus Chirag is at Position 4, leaving Eshan at Position 3.

Step 4: Analyzing Round 4 and Round 5
In Round 4, Diya (Position 6) passes to an intermediate child at position $P_4$, who then passes to Aarav (Position 2) in Round 5.
For this path to be valid, $P_4$ must be a neighbor (within 2 positions) of both 6 and 2. The only positions that satisfy this are:
• Position 1 (Bina): since 6 to 1 is Second to the left, and 1 to 2 is Immediately to the left.
• Position 7 (whomever is seated there): since 6 to 7 is Immediately to the left, and 7 to 2 is Second to the left.

Step 5: Counting Buck Receptions for Gaurav
Let us verify how many times Gaurav receives the buck across all valid seating configurations and path choices:
• Gaurav is uniquely fixed at Position 5.
• Looking at the receivers of the Buck in each round:
  – Round 1: Aarav (Position 2)
  – Round 2: Position 7
  – Round 3: Diya (Position 6)
  – Round 4: Position 1 or Position 7
  – Round 5: Aarav (Position 2)
  – Round 6: Position 4
  – Round 7: Gaurav (Position 5)
  – Round 8: Position 6 (Diya)
  – Round 9: Farhan
  – Round 10: Chirag

Regardless of which child is seated at Position 7, 4, or 3, and regardless of whether the Round 4 receiver is Bina (Position 1) or the child at Position 7, Gaurav receives the buck exactly 1 time (in Round 7).
For other options like Farhan, the number of times they receive the buck varies between 2 and 3 depending on the seating arrangement and choice of path.
Thus, the only child for whom we can determine the exact number of buck receptions is Gaurav.