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 pass types can the total number of occurrences be uniquely determined?

Correct Answer: Immediately to the right

To find which pass type's total occurrences can be uniquely determined, we first need to reconstruct the circular seating arrangement of the seven children and track the movement of the Buck in each round.

Step 1: Determine the Seating Arrangement
Let us label the 7 positions on the circle clockwise as 1, 2, 3, 4, 5, 6, and 7, where the children face inwards. Facing inwards means:
• Passing Immediately to the left is a clockwise move of +1 position.
• Passing Second to the left is a clockwise move of +2 positions.
• Passing Immediately to the right is a counter-clockwise move of -1 position.
• Passing Second to the right is a counter-clockwise move of -2 positions.

From the first image, the initial positions are:
• Position 1: Bina
• Position 2: Aarav
• Position 6: Diya

Let's trace the rounds using the table information:
• The game starts with Bina at Position 1.
• Round 1: Bina passes Immediately to the left (+1) to Aarav. Aarav is at position:
$$1+1=2$$
This matches the diagram where Aarav is at Position 2.
• Round 2: Aarav (Position 2) passes Second to the right (-2). The recipient is at position:
$$2-2=0\equiv 7\left(\mathrm{mod}7\right)$$
From the third image, we see that Eshan is at Position 7. So, Eshan receives the Buck in Round 2.
• Round 3: Eshan (Position 7) passes Immediately to the right (-1) to Diya. Diya is at position:
$$7-1=6$$
This matches the diagram where Diya is at Position 6.
• Round 7: The Buck is passed Immediately to the left (+1) to Gaurav. From the second and third images, Gaurav is at Position 5. Thus, the sender in Round 7 must have been at position:
$$5-1=4$$
From the third image, Farhan is at Position 4. So the Buck was passed from Farhan (Position 4) to Gaurav (Position 5) in Round 7.
• Round 6: The pass type is Second to the left (+2) and must land on Farhan (Position 4). Thus, the sender in Round 6 must have been at position:
$$4-2=2$$
Position 2 is Aarav. So, Aarav is the receiver of Round 5 and sender of Round 6.

The complete clockwise seating arrangement from Position 1 to 7 is:
1: Bina, 2: Aarav, 3: Chirag, 4: Farhan, 5: Gaurav, 6: Diya, 7: Eshan

Step 2: Trace the Paths and Fill in the Missing Moves
Let us write down the positions at the end of each round:
• Start: Bina (Position 1)
• Round 1: Aarav (Position 2) — Pass type: +1 (Immediately to the left)
• Round 2: Eshan (Position 7) — Pass type: -2 (Second to the right)
• Round 3: Diya (Position 6) — Pass type: -1 (Immediately to the right)
• Round 4: [Unknown Position]
• Round 5: Aarav (Position 2)
• Round 6: Farhan (Position 4) — Pass type: +2 (Second to the left)
• Round 7: Gaurav (Position 5) — Pass type: +1 (Immediately to the left)
• Round 8: [Unknown Position] — Pass type: +1 (Immediately to the left) from Gaurav (Position 5). Thus, the receiver is:
$$5+1=6\left(\text{Diya}\right)$$
• Round 9: Farhan (Position 4). Since the sender was Diya (Position 6), the transition is:
$$4-6=-2\left(\text{Second to the right}\right)$$
• Round 10: Chirag (Position 3). Since the sender was Farhan (Position 4), the transition is:
$$3-4=-1\left(\text{Immediately to the right}\right)$$

Step 3: Analyze the Undetermined Steps (Rounds 4 and 5)
The Buck moves from Diya (Position 6) at the end of Round 3 to Aarav (Position 2) at the end of Round 5 in two steps:
$$6+x_4+x_5\equiv 2\left(\mathrm{mod}7\right)\Rightarrow x_4+x_5\equiv -4\equiv 3\left(\mathrm{mod}7\right)$$
Where the pass values $x_4,x_5\in \{+1,-1,+2,-2\}$.
The possible pairs for $(x_4,x_5)$ that sum to 3 (mod 7) are:
1. $(+1,+2)$
2. $(+2,+1)$
3. $(-2,-2)$

Let us count the occurrences of each pass type in all 10 rounds based on these three cases:

As shown in the table, the total number of occurrences of the pass type "Immediately to the right" is always exactly 2 in all possible cases, which makes it uniquely determined.