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.

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

To find the bids of the four players—Arun, Bankim, Charu, and Dipak—in the first round, let us analyze the scoring rules of the Hi-Lo game and the given facts step-by-step.

First, we note the points awarded to players depending on the number of players bidding Hi in a round:
1. 4 Hi, 0 Lo: All players lose 1 point each (individual score: -1).
2. 3 Hi, 1 Lo: The three Hi bidders gain 1 point each; the Lo bidder loses 3 points.
3. 2 Hi, 2 Lo: The two Hi bidders gain 2 points each; the two Lo bidders lose 2 points each.
4. 1 Hi, 3 Lo: The single Hi bidder gains 3 points; the three Lo bidders lose 1 point each.
5. 0 Hi, 4 Lo: All players gain 1 point each.

Let us analyze the cumulative scores at the end of the first three rounds (Fact 1):
• Arun: 6 points
• Dipak: 2 points
• Bankim: -2 points
• Charu: -2 points

Since the maximum points a player can score in a single round is 3 (when they are the only player bidding Hi), the only way for Arun to achieve a cumulative score of 6 points over three rounds without scoring 0 in any round (as 0 is not a possible single-round score for any bidding outcome) is to score:
$$3+2+1=6$$

This means that in the first three rounds, Arun's individual round scores must be 3, 2, and 1 in some order:
• The round where Arun scored 3 points must be a 1 Hi, 3 Lo round (where Arun bid Hi and the other three bid Lo).
• The round where Arun scored 2 points must be a 2 Hi, 2 Lo round (where Arun and one other player bid Hi, scoring 2 points each, and the other two players bid Lo, scoring -2 points each).
• The round where Arun scored 1 point must be a 0 Hi, 4 Lo round (where all four players bid Lo and scored 1 point each).

Now, let us determine Dipak's scores in these three rounds. In the three rounds, the scores for the players other than Arun are:
• In the 1 Hi (Arun) round: -1 point each.
• In the 2 Hi (Arun + one other) round: +2 points for the other Hi bidder, and -2 points each for the two Lo bidders.
• In the 0 Hi round: +1 point each.
Since Dipak has a cumulative score of 2 points at the end of three rounds, he must have been the other player who bid Hi in the 2 Hi round. Thus, Dipak's round scores must be a permutation of:
$$\{2,-1,1\}$$

According to Fact 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:
$$\text{Score in Round 2}<\text{Score in Round 3}<\text{Score in Round 1}$$
Arranging Dipak's three scores $\{-1,1,2\}$ in this order gives:
• Score in Round 1 = 2
• Score in Round 2 = -1
• Score in Round 3 = 1

Since Dipak scored 2 points in the first round, the first round must be the 2 Hi, 2 Lo round, where the two players bidding Hi were Arun and Dipak, and the two players bidding Lo were Bankim and Charu.

Therefore, the bids in the first round by Arun, Bankim, Charu, and Dipak, respectively, were:
Hi, Lo, Lo, Hi