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:

What were the bids by Arun, Bankim, Charu and Dipak, respectively in the first round?

From the given information, we can see that the points scored by the players in a round has the following possibilities:

HHHH: (-1, -1, -1, -1)

HHHL: (1, 1, 1, -3)

HHLL: (2, 2, -2, -2)

HLLL: (3, -1, -1, -1)

LLLL: (1, 1, 1, 1)

Also, the total points scored by the four players in a round can only be -4 or 0 or 4.

From (1), the total points scored by the four players combined in the first three rounds is 6 + 2 - 2 - 2 = 4.

Hence, in the first three rounds, the total points scored by the four players must be either (-4, 4, 4) OR (0, 0, 4), in any order.

Also, from (1), in the first three rounds, Arun scored 6 points. And from (2), Arun scored 7 points at the end of round 6. Hence, in the 4th , 5th and 6th rounds, he must have scored 1 point.

From (4), Arun scored 3 points in exactly 2 rounds.

These two rounds cannot both be among 4th , 5th and 6th rounds because he scored a net of only 1 point in these three rounds combined.

Hence, Arun must have scored 3 points in one round among 1st , 2nd and 3 rd rounds. If Arun scored 3 points in the first three rounds, then in that round, the total points scored by the four players combined must be 0 (in the case of HLLL).

Hence, the total points scored by the four players in the first three rounds must be (0, 0, 4). Among the first three rounds, in one round, the three players must have scored (3, -1, -1, -1), with Arun scoring 3 points.

Since in another round, the four players scored a total of 4 points, they must have bid LLLL (as it is the only case in which they can score 4 points in total). They must have scored (1, 1, 1, 1).

Since Arun scored a total of 6 points in the first three rounds, and he scored 3 points and 1 point in two of these rounds, he must have score 2 points in the other round. This is possible only if the players bid HHLL and the scores of the four players must be (2, 2, - 2, - 2).

In the round that the players scored (3, - 1, - 1, - 1), Dipak must have scored - 1 points (since Arun scored 3 points).

In the round that the players scored (1, 1, 1, 1), Dipak must have scored 1 point.

In the round that the players scored (2, 2, - 2, - 2), Dipak must have scored 2 points (since the total points that Dipak scored in the first three rounds is 2).

From (3), Dipak must have scored 2 points in the first round, - 1 points in the second round and 1 point in the third round.

From this, we can fill the points for the first three rounds, as shown below

(Note that with this information, the first question of the set can be answered)

In the next three rounds, from (1) and (2), Arun must have scored 1 point, Bankim must have scored 1 point, Charu must have scored - 3 points and Dipak must have scored - 3 points.

The total points scored by the four players are - 4. This is possible if the total points scored by the four players in the three rounds are (0, 0, - 4) OR (4, - 4, - 4) in any order.

However, we know that Arun must have scored 3 points in one of these three rounds (from (4)). Hence, the total points scored by the players in this round must be 0.

Hence, the four players must have scored (0, 0, - 4) points in these three rounds. In one round the points scored by the players must be (3, - 1, - 1, - 1), with Arun scoring 3 points.

In the round in which the total points scored by the four players is 4, they must have scored ( - 1, - 1, - 1, - 1).

Since Bankim scored a total of 1 point in these three rounds, and he scored - 1 point and - 1 point in the two rounds mentioned above, he must have scored 3 points in the other third.

Hence, in the remaining round, the four players must have scored (3, - 1, - 1, - 1), with Bankim scoring 3 points.

However, with the given information, we cannot deduce the round number corresponding to the above rounds.

Hence, we get the following table:

**Question 1:**
The bids by Arun, Bankim, Charu, Dipak in the first round were Hi, Lo, Lo, Hi.

Option: 4

**Question 2:**
Arun bid Hi in 4 rounds

Answer: 4

**Question 3:**
Bankim bid Lo in 4 rounds

Answer: 4

**Question 4:**
All four players made identical bids in 2 rounds

Answer: 2

**Question 5:**
Dipak gained exactly 1 point in 1 round

Answer: 1

**Question 6:**
In the second round, Arun was the only player to bid Hi.

Option: 4