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 how many rounds did all four players make identical bids?

The correct option/answer is 2.

To understand why this is the correct answer, let us break down the points system and analyze the rounds step-by-step.

1. Point System Analysis:
Let us analyze the possible points awarded to each player depending on the number of players bidding Hi or Lo in any round:

• Case 1 (4 Hi, 0 Lo): All four players bid Hi. Each player gets $-1$ point. The sum of scores for this round is $-4$ points.
• Case 2 (3 Hi, 1 Lo): Three players bid Hi (getting $+1$ point each), one bids Lo (getting $-3$ points). The sum of scores is $3\left(1\right)+1(-3)=0$.
• Case 3 (2 Hi, 2 Lo): Two players bid Hi (getting $+2$ points each), two bid Lo (getting $-2$ points each). The sum of scores is $2\left(2\right)+2(-2)=0$.
• Case 4 (1 Hi, 3 Lo): One player bids Hi (getting $+3$ points), three bid Lo (getting $-1$ point each). The sum of scores is $1\left(3\right)+3(-1)=0$.
• Case 5 (0 Hi, 4 Lo): All four players bid Lo. Each player gets $+1$ point. The sum of scores for this round is $+4$ points.

2. Analyzing the First Three Rounds:
At the end of three rounds, the scores are:
Arun = 6, Dipak = 2, Bankim = -2, Charu = -2.
The sum of all players' scores after 3 rounds is:

$6+2+(-2)+(-2)=4$

Since the round sums can only be $-4$, $0$, or $4$, the only way to obtain a total sum of $4$ over three rounds is to have:

• One round where everyone bids Lo (Case 5, sum = $+4$, where everyone gets $+1$). This is our first round of identical bids.
• Two mixed rounds (sum = 0 for each).

Since one of the rounds was 4 Lo (all four getting $+1$), Arun must have scored $6-1=5$ points from the remaining two mixed rounds. The only possible way for a player to score $5$ points from two mixed rounds is by scoring $+3$ in one round (Case 4: only Arun bids Hi) and $+2$ in the other round (Case 3: two players bid Hi, one of whom is Arun).
Thus, the first three rounds consist of:

1. One round of 4 Lo (all four players bid Lo; scores: A = 1, B = 1, C = 1, D = 1).
2. One round of 1 Hi, 3 Lo with Arun bidding Hi (scores: A = 3, B = -1, C = -1, D = -1).
3. One round of 2 Hi, 2 Lo with Arun and Dipak bidding Hi (scores: A = 2, B = -2, C = -2, D = 2).

This gives the required scores after 3 rounds: A = 6, D = 2, B = -2, C = -2.

3. Determining the Round Sequence for the First Three Rounds:
According to Fact 3, Dipak's score in the third round was less than his score in the first round but more than his score in the second round. Dipak's individual scores in the three rounds are $+2$, $-1$, and $+1$.
Ordering these scores: $-1<+1<+2$.
Therefore, Dipak's scores in rounds 1, 2, and 3 must be:

• Round 1 score = $+2$ (the 2 Hi, 2 Lo round)
• Round 2 score = $-1$ (the 1 Hi, 3 Lo round with Arun bidding Hi)
• Round 3 score = $+1$ (the 4 Lo round)

This confirms that Round 3 is the round where all four players bid Lo (identical bids).

4. Analyzing the Last Three Rounds (Rounds 4 to 6):
At the end of six rounds, the scores are:
Arun = 7, Bankim = -1, Charu = -5, Dipak = -1.
The sum of all scores is:

$7+(-1)+(-5)+(-1)=0$

Since the sum after the first three rounds was $4$, the sum of scores for rounds 4, 5, and 6 must be $0-4=-4$. The only way to get a sum of $-4$ over three rounds is:

• One round where everyone bids Hi (Case 1, sum = $-4$, where everyone gets $-1$). This is our second round of identical bids.
• Two mixed rounds (sum = 0 for each).

Fact 4 states that Arun was the only player who bid Hi in exactly two out of the six rounds. Since one such round occurred in the first three rounds, the second such round must occur in the last three rounds.
In this round of 1 Hi, 3 Lo with Arun bidding Hi: Arun gets $+3$, and the others get $-1$ each.
Subtracting the points from the 4 Hi round (everyone gets $-1$) and this second 1 Hi round (Arun gets $+3$, others get $-1$) from the change in scores from Round 3 to Round 6:

• Arun's change: $7-6=1$ point. Excluding the 4 Hi round ($-1$) and the 1 Hi round ($+3$), Arun scores: $1-(-1)-3=-1$ point in the remaining mixed round.
• Bankim's change: $-1-(-2)=1$ point. Excluding the 4 Hi round ($-1$) and the 1 Hi round ($-1$), Bankim scores: $1-(-1)-(-1)=3$ points in the remaining mixed round.
• Charu's change: $-5-(-2)=-3$ points. Excluding the two rounds, Charu scores: $-3-(-1)-(-1)=-1$ point in the remaining mixed round.
• Dipak's change: $-1-2=-3$ points. Excluding the two rounds, Dipak scores: $-3-(-1)-(-1)=-1$ point in the remaining mixed round.

In the last remaining mixed round, Bankim scores $+3$ and all other three players score $-1$, which corresponds to a round of 1 Hi, 3 Lo with Bankim bidding Hi.

Conclusion:
The rounds played by the four players across the six rounds are:

1. Round 1: Mixed bids (2 Hi, 2 Lo)
2. Round 2: Mixed bids (1 Hi, 3 Lo)
3. Round 3: All four bid Lo (Identical bids #1)
4. Round 4: All four bid Hi (Identical bids #2)
5. Round 5: Mixed bids (1 Hi, 3 Lo)
6. Round 6: Mixed bids (1 Hi, 3 Lo)

Thus, all four players made identical bids in exactly 2 rounds.