The Hi-Lo game is a four-player game played in six rounds. In every round, each player chooses to bid Hi or Lo. The bids are made simultaneously. If all four bid Hi, then all four lose 1 point each. If three players bid Hi and one bids Lo, then the players bidding Hi gain 1 point each and the player bidding Lo loses 3 points. If two players bid Hi and two bid Lo, then the players bidding Hi gain 2 points each and the players bidding Lo lose 2 points each. If one player bids Hi and three bid Lo, then the player bidding Hi gains 3 points and the players bidding Lo lose 1 point each. If all four bid Lo, then all four gain 1 point each.

Four players Arun, Bankim, Charu, and Dipak played the Hi-Lo game. The following facts are known about their game:

1. At the end of three rounds, Arun had scored 6 points, Dipak had scored 2 points, Bankim and Charu had scored -2 points each.
2. At the end of six rounds, Arun had scored 7 points, Bankim and Dipak had scored -1 point each, and Charu had scored -5 points.
3. Dipak’s score in the third round was less than his score in the first round but was more than his score in the second round.
4. In exactly two out of the six rounds, Arun was the only player who bid Hi.

In which of the following rounds, was Arun DEFINITELY the only player to bid Hi?

The correct option is Second.

Let us break down the logical reasoning step-by-step to understand why Arun was definitely the only player to bid Hi in the second round.

Step 1: Understand the scoring rules per round
Let $k$ be the number of players who bid Hi in a given round. The points scored by players in that round are as follows:
• If $k=4$ (All 4 bid Hi): Every player gets $-1$ point. (Total round score = $-4$)
• If $k=3$ (3 bid Hi, 1 bids Lo): Hi players get $+1$, Lo player gets $-3$. (Total round score = $3\left(1\right)-3=0$)
• If $k=2$ (2 bid Hi, 2 bid Lo): Hi players get $+2$, Lo players get $-2$. (Total round score = $2\left(2\right)+2(-2)=0$)
• If $k=1$ (1 bids Hi, 3 bid Lo): Hi player gets $+3$, Lo players get $-1$. (Total round score = $1\left(3\right)+3(-1)=0$)
• If $k=0$ (All 4 bid Lo): Every player gets $+1$ point. (Total round score = $+4$)

Step 2: Analyze the scores after the first three rounds
At the end of three rounds, the scores are:
• Arun (A): $6$
• Dipak (D): $2$
• Bankim (B): $-2$
• Charu (C): $-2$

The sum of all four players' scores after three rounds is:
$6+2+(-2)+(-2)=4$

Since the total sum of scores in any round is either $-4$, $0$, or $+4$, the only way three rounds can sum to $4$ is if:
• One round has a total score of $+4$ (where $k=0$, so all four players bid Lo and get $+1$ point each).
• The other two rounds have a total score of $0$ each (where $k\in ;\{1,2,3\}$).

Step 3: Determine the outcomes of these three rounds
In the round where everyone bid Lo ($k=0$), Arun receives $+1$ point. Since his total score is $6$, his scores in the other two rounds must sum to:
$6-1=5$

The only possible positive scores a player can obtain in a round with a total score of $0$ are $+1$ (if $k=3$), $+2$ (if $k=2$), and $+3$ (if $k=1$).
To sum to $5$ over two rounds, Arun must have scored:
• $+3$ in one round (meaning $k=1$; Arun was the only player who bid Hi).
• $+2$ in another round (meaning $k=2$; Arun and one other player bid Hi).

Thus, the individual scores for the three rounds are:
• Round type 1 (All Lo): Every player gets $+1$.
• Round type 2 (Arun only Hi): Arun gets $+3$, others get $-1$ each.
• Round type 3 (Arun and player P bid Hi): Arun and P get $+2$ each, others get $-2$ each.

For Dipak to end up with $2$ points after these three rounds:
• From Round type 1, Dipak gets $+1$.
• From Round type 2, Dipak gets $-1$.
• Therefore, in Round type 3, Dipak must have scored $+2$ (which means Dipak is player P, bidding Hi alongside Arun).
This perfectly matches the remaining players Bankim and Charu scoring $-2$ each (getting $1-1-2=-2$).

Step 4: Establish the round sequence using Dipak's scores
From Fact 3, Dipak's score in the third round was less than in the first round but more than in the second round:
$D_2<D_3<D_1$

Dipak's scores in the three rounds are $+2$, $-1$, and $+1$. Ordering these values gives:
$-1<+1<+2$

Comparing this to the inequality, we get:
• First Round: Dipak scored $+2$ (Arun and Dipak bid Hi).
• Second Round: Dipak scored $-1$ (Arun was the only player who bid Hi).
• Third Round: Dipak scored $+1$ (All players bid Lo).

Therefore, Arun was definitely the only player to bid Hi in the Second round.