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. Dipakâ€™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.

- Hi, Lo, Lo, Lo
- Hi, Hi, Lo, Lo
- Lo, Lo, Lo, Hi
- Hi, Lo, Lo, Hi

- Third
- First
- Second
- Fourth

From point 3) it is clear that in first three rounds Dipak scored highest point in round 1. Which must be positive so option 1) and option 4) can not be true. Now if Arun score negative points in first round , his cumulative points can not be 6 after 3 round . So option 2 is incorrect as well.

Thus correct option 3)

The points given in after each round can be tabulated as below:

Round | Arun | Bankim | Charu | Dipak |

1 | ||||

2 | ||||

3 | 6 | -2 | -2 | 2 |

4 | ||||

5 | ||||

6 | 7 | -1 | -5 | -1 |

Arun has 6 points after round 3 it means his points in these three rounds must have been 1,2 and 3 in any order . (There is no other possibility )

From point 3) it is clear that in first three rounds Dipak scored highest point in round 1. Which must be positive thus both Arun and Dipak bid Hi and other two bid low in first round.

First Round Bid : Hi, Lo, Lo, Hi (Arun 2, Bankim -2 , Charu -2 and Dipak + 2)

Now Dipak must score negative in 2nd bid as he had scored least point in 2nd round if compare first three rounds .

Second Round Bid : Hi, Lo, Lo, Lo (Arun 3, Bankim -1 , Charu -1 and Dipak -1)

Now we can see from the points after third round that each player gain 1 points while in third round so

Third Round Bid : Lo, Lo, Lo, Lo (Arun 1, Bankim 1 , Charu 1 and Dipak 1)

From point 4) it is clear that Arun must have scored 3 points in any one of round 4, 5 or 6 . Thus in 4 rounds his cumulative points = 6+3 = 9

Now as his final score after 6 rounds is 7 so there is two possibility of his scores in other two rounds

Case 1 â€“ If he scores + 1 and -3

Next Three bid will be in order of (Arun, Bankim, Charu, Dipak)

Hi, Lo, Lo, Lo,

Lo , Hi , Hi , Hi ,

Lo , Lo , Lo , Lo or

Hi , Hi , Lo Hi

If we see the scores after 6th round This case is not possible as Dipak will have 3 points but it should have -1 points .

Case 2 : if he scores -1 , -1

Next Three bid will be in order of (Arun, Bankim, Charu, Dipak)

Hi, Lo, Lo, Lo

Lo , Hi , Lo , Lo

Hi , Hi , Hi Hi

Which is the possible bid as per the given scores after six round.

Now we can answer all the questions .

Thus Final Table for each round bids will look like

Round | Arun | Bankim | Charu | Dipak |

1 | Hi | Lo | Lo | Hi |

2 | Hi | Lo | Lo | Lo |

3 | Lo | Lo | Lo | Lo |

4/5/6 | Hi | Lo | Lo | Lo |

4/5/6 | Lo | Hi | Lo | Lo |

4/5/6 | Hi | Hi | Hi | Hi |

Q1. Option c) correct

Q2. Correct answer 4. ( 2 Hi in first 3 rounds and 2 Hi in last three Rounds)

Q3. Correct answer 4 ( 3 Lo in first 3 rounds and 1 Lo in last three rounds)

Q4. Two rounds (Third round and one among 4th , 5th and 6th)

Q5. Deepak gained exactly one point in third round only .

Q6. From the above solution we can say there were only two rounds in which there were only one Hi bid one was second round and other was one among fourth , fifth and sixth round.

So option 1) Second will be correct as it was the only definite round in which there was only one Hi bids.

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