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 Arun bid Hi?

The correct answer is 4.

Step 1: Understand the Scoring System
In each round, the four players choose either Hi or Lo. Let $H$ be the number of players bidding Hi, and $L$ be the number of players bidding Lo ($H+L=4$). The scores gained by the players in each scenario are as follows:
1. If $H=4$ (all bid Hi): All four players lose 1 point each. The sum of the scores in this round is $4\times (-1)=-4$ points.
2. If $H=3,L=1$: Players bidding Hi gain 1 point each, and the player bidding Lo loses 3 points. The sum of the scores is $3\times 1+1\times (-3)=0$ points.
3. If $H=2,L=2$: Players bidding Hi gain 2 points each, and players bidding Lo lose 2 points each. The sum of the scores is $2\times 2+2\times (-2)=0$ points.
4. If $H=1,L=3$: The player bidding Hi gains 3 points, and players bidding Lo lose 1 point each. The sum of the scores is $1\times 3+3\times (-1)=0$ points.
5. If $H=0$ (all bid Lo): All four players gain 1 point each. The sum of the scores is $4\times 1=+4$ points.

Step 2: Analyze the Sum of Scores after Rounds 3 and 6
At the end of three rounds, the scores are:
Arun = 6 points, Dipak = 2 points, Bankim = -2 points, Charu = -2 points.
The sum of the players' scores after three rounds is:
$6+2+(-2)+(-2)=4$
Since a single round can only result in a sum of -4, 0, or 4 points, the only way to obtain a sum of 4 points in three rounds is if there is exactly one round with a sum of 4 (where $H=0$, all bid Lo) and two rounds with a sum of 0.

At the end of six rounds, the scores are:
Arun = 7 points, Bankim = -1 point, Dipak = -1 point, Charu = -5 points.
The sum of the players' scores after six rounds is:
$7+(-1)+(-1)+(-5)=0$
Since the total sum after three rounds was 4, the sum of scores in the last three rounds (rounds 4 to 6) must be $0-4=-4$ points. The only way to get a sum of -4 points in three rounds is if there is exactly one round with a sum of -4 (where $H=4$, all bid Hi) and two rounds with a sum of 0.

Thus, across the six rounds, the types of rounds played were:
- 1 round of $H=0$ (every player gets +1 point)
- 1 round of $H=4$ (every player gets -1 point)
- 4 rounds of sum 0 (where $H\in \{1,2,3\}$)

Step 3: Analyze Arun's Score and Bids
From Fact 4, in exactly two of the six rounds, Arun was the only player who bid Hi (meaning $H=1$ with Arun bidding Hi). In each of these two rounds, Arun scored +3 points, giving him a total of +6 points from these rounds.
Let us calculate Arun's score from the known rounds so far:
- From the single $H=0$ round (all bid Lo): Arun gets +1 point.
- From the single $H=4$ round (all bid Hi): Arun gets -1 point.
- From the two rounds where Arun is the sole Hi bidder: Arun gets $2\times 3=6$ points.
The sum of Arun's scores from these four rounds is $1-1+6=6$ points.

Since Arun's total score after six rounds is 7 points, he must have scored a total of $7-6=1$ point in the remaining two rounds (both of which are sum 0 rounds where Arun is NOT the sole Hi bidder).
The possible scores a player can get in a sum 0 round (excluding being the sole Hi bidder) are:
- +1 point (bidding Hi when $H=3$)
- -3 points (bidding Lo when $H=3$)
- +2 points (bidding Hi when $H=2$)
- -2 points (bidding Lo when $H=2$)
- -1 point (bidding Lo when $H=1$)

To get a sum of +1 point over these two rounds, the only possible combination of scores is:
- +2 points (bidding Hi in an $H=2$ round)
- -1 point (bidding Lo in an $H=1$ round)

Step 4: Count the Number of Rounds Arun Bid Hi
Now we list the bids of Arun in each of the 6 rounds:
1. In the $H=0$ round: Arun bid Lo (0)
2. In the $H=4$ round: Arun bid Hi (1)
3. In the two rounds where Arun was the sole Hi bidder: Arun bid Hi in both (2)
4. In the remaining two rounds: Arun bid Hi in the round where he scored +2 points, and bid Lo in the round where he scored -1 point (1)

Adding these together, the number of rounds in which Arun bid Hi is:
$1+2+1=4$
Therefore, Arun bid Hi in exactly 4 rounds.