Question: What were the bids by Arun, Bankim, Charu and Dipak, respectively in the first round?
Explanation:
Points scored in a round has the following possibilities:
H H H H: (-1, -1, -1, -1); Total -4
Total points scored in a round can be -4, 0 or 4.
We can make the following table from the data given.
Net score after first 3 rounds is 6 – 2 – 2 + 2 = 4.
This could have happened if the total score in first three rounds was {4, 4, -4} or {4, 0, 0} in any order.
∴ In one of the first 3 rounds the net score has to be 4. This is only possible when all of them bid Lo and score 1 point each.
Now, Arun scored total 6 points in first 3 rounds. Also, he scores 1 in one of the first 3 rounds. This is possible if he scores {3, 2, 1} in any order in the first 3 rounds.
For Arun to score 3 in a round he will have to bid Hi and other will bid Lo. Other 3 will score -1 each.
For Arun to score 2 in a round he will have to bid Hi along with one other and the remaining two will bid Lo. Scores of other 3 will be {2, -2, -2} in any order.
Dipak’s score in two of the rounds is 1 and -1. His net score in 3 rounds is 2. Hence, Dipak scores 2 in one of the first three rounds.
Also, 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.
∴ We can make the following table for the first 3 rounds.
Net score in last 3 rounds is 1 + 1 – 3 - 3 = -4.
This could have happened if the total score in last three rounds was {4, -4, -4} or {-4, 0, 0} in any order.
∴ In one of the last 3 rounds the net score has to be -4. This is only possible when all of them bid Hi and score -1 point each.
In exactly two out of the six rounds, Arun was the only player who bid Hi i.e., Arun scores 3 in exactly 2 of the six rounds.
∴ Arun scored 3 in one of the last 3 rounds. Also, his net score in last 3 rounds is 1. Hence, he scores -1 and -1 in 2 of the last 3 rounds.
∴ We can make the following table for the all 6 rounds.
In first round Arun, Bankim, Charu and Dipak bid Hi, Lo, Lo and Hi.
Hence, option (c).