Question

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?[This Question was asked as TITA]

• A. 5 rounds
• B. 3 rounds
• C. 1 rounds
• D. 2 rounds

Answer 1

The Correct Option is D

Given:

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.

Let's analyze the information:

• Arun scored 1 point between the third and sixth rounds, so one of the rounds where he was the only one bidding Hi must be the sixth round.
• Arun bid Hi alone in two rounds, meaning he lost 1 point in these two rounds. If one of them was the sixth round, then the other one could be the second, third, or fourth round.

Let's assume Arun bid Hi alone in the second and sixth rounds:

• Arun bid Hi alone in the second round, and his score after the third round was 6 points. Therefore, his score after the second round was 6 - 3 = 3 points.
• 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. Since his score after the third round was 2 points, his score after the first round must have been 2 - 3 = -1 points. Therefore, his score after the second round was -1 + 3 = 2 points. This implies that Dipak bid Hi in the first round and Lo in the second and third rounds.
• Bankim and Charu both scored -2 points after the third round, which means they must have bid Hi in the first round and Lo in the second and third rounds.

Now, let's analyze Bankim and Dipak's scores after the sixth round:

• Bankim and Dipak both scored -1 point after the sixth round. For them to have the same score, Bankim must have bid Hi in the fourth round, and Dipak must have bid Lo in the fourth round.

Considering this, Bankim and Dipak both bid Hi in the first round and Lo in the second and fourth rounds.

Now, let's find the round where all four players made identical bids:

• In the first round, all four players bid Hi.
• In the second and fourth rounds, all four players bid Lo.

So, all four players made identical bids in 2 rounds.

Therefore, the answer is 2 rounds.