Mindyard | 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 Bankim bid Lo?

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 Bankim bid Lo?

Correct Answer :

Solution :

The correct answer is 4.

To find the number of rounds in which Bankim bid Lo, we can analyze the game scoring rules and the players' scores step-by-step.

Step 1: Analyze the Scoring Rules and Round Sums

Let _$k$ be the number of players bidding Hi in a round. The points gained or lost by the players and the sum of all players' scores in that round are:

If $k = 4$ (4 Hi, 0 Lo): All four lose 1 point each.
Round Sum = $4 \times (- 1) = - 4$ .
If $k = 3$ (3 Hi, 1 Lo): Hi players gain 1 point, Lo player loses 3 points.
Round Sum = $3 \times 1 + 1 \times (- 3) = 0$ .
If $k = 2$ (2 Hi, 2 Lo): Hi players gain 2 points, Lo players lose 2 points.
Round Sum = $2 \times 2 + 2 \times (- 2) = 0$ .
If $k = 1$ (1 Hi, 3 Lo): Hi player gains 3 points, Lo players lose 1 point.
Round Sum = $1 \times 3 + 3 \times (- 1) = 0$ .
If $k = 0$ (0 Hi, 4 Lo): All four players gain 1 point.
Round Sum = $4 \times 1 = 4$ .

Step 2: Analyze the First Three Rounds (R1, R2, R3)

At the end of three rounds:
Arun = 6 points, Dipak = 2 points, Bankim = -2 points, Charu = -2 points.
The sum of all scores after 3 rounds is:

$6 + 2 + (- 2) + (- 2) = 4$

Since the round sums can only be -4, 0, or 4, the only way to get a total sum of 4 in 3 rounds is to have:

One round with $k = 0$ (round sum = 4, where everyone gets +1)
Two rounds with $k \in {1, 2, 3}$ (round sums = 0)

In the $k = 0$ round, Arun scores +1. Thus, in the other two rounds, Arun must score a total of $6 - 1 = 5$ points. Since the maximum score a player can obtain in a single round is +3, Arun's score in these two rounds must be +3 and +2.
• An Arun score of +3 implies a $k = 1$ round where Arun is the sole Hi bidder.
• An Arun score of +2 implies a $k = 2$ round where Arun is one of the two Hi bidders.

Let's determine who the other Hi bidder is in the $k = 2$ round:
Dipak's total score in these three rounds is 2.
• In the $k = 0$ round, Dipak gets +1.
• In the $k = 1$ round (where Arun is the only Hi bidder), Dipak gets -1.
• To reach a total of 2 points, Dipak must score $2 - (1 - 1) = 2$ points in the $k = 2$ round.
This means Dipak is indeed the other player who bid Hi in the $k = 2$ round.

Thus, the individual scores for the three rounds are:
• $k = 0$ round: Arun (+1), Bankim (+1), Charu (+1), Dipak (+1)
• $k = 1$ round (only Arun is Hi): Arun (+3), Bankim (-1), Charu (-1), Dipak (-1)
• $k = 2$ round (Arun & Dipak are Hi): Arun (+2), Bankim (-2), Charu (-2), Dipak (+2)

According to Fact 3, Dipak's score in the third round was less than in the first round but more than in the second round. Ordering Dipak's round scores (+2, -1, +1) gives:
• Round 1: +2 (Dipak bid Hi, Bankim bid Lo)
• Round 2: -1 (Dipak bid Lo, Bankim bid Lo)
• Round 3: +1 (Dipak bid Lo, Bankim bid Lo)
Consequently, in the first 3 rounds, Bankim bid Lo in all 3 rounds.

Step 3: Analyze the Last Three Rounds (R4, R5, R6)

At the end of six rounds, the final scores are:
Arun = 7, Bankim = -1, Charu = -5, Dipak = -1.
Subtracting the scores from the end of Round 3, the change in scores over the last 3 rounds is:
• Arun: $7 - 6 = 1$
• Bankim: $- 1 - (- 2) = 1$
• Charu: $- 5 - (- 2) = - 3$
• Dipak: $- 1 - 2 = - 3$

The total sum of points in these rounds is $1 + 1 + (- 3) + (- 3) = - 4$ .
Thus, the configuration of the last three rounds must consist of:

One round with $k = 4$ (round sum = -4, everyone gets -1)
Two rounds with $k \in {1, 2, 3}$ (round sums = 0)

Fact 4 states that Arun was the only player who bid Hi in exactly two rounds. Since this happened once in the first 3 rounds (Round 2), it must happen exactly once in the last 3 rounds.
Let this round (where Arun is the only Hi player) be R_x.
• In R_x, Arun scores +3, and the others (including Bankim) score -1.
• In the $k = 4$ round, everyone (including Bankim) scores -1.
For Arun to achieve a total change of +1 across the three rounds:
$Score in R_y = 1 - (- 1 [from k = 4]) - (3 [from R_x]) = - 1$ .
Since R_y has a sum of 0, and Arun scored -1, Arun must have bid Lo.
To maintain the net scores (Bankim = +1, Charu = -3, Dipak = -3):
• Bankim must score +3 in R_y (meaning Bankim was the only player who bid Hi in R_y).
• Charu and Dipak score -1 in R_y.
This satisfies all player net scores:
• Bankim's score change: -1 (from $k = 4$ ) - 1 (from R_x) + 3 (from R_y) = 1.
• Charu's score change: -1 - 1 - 1 = -3.
• Dipak's score change: -1 - 1 - 1 = -3.

Let's find Bankim's bids in the last three rounds (R4, R5, R6):
• In the $k = 4$ round: Bankim bid Hi.
• In the R_x round (Arun only Hi): Bankim bid Lo.
• In the R_y round (Bankim only Hi): Bankim bid Hi.
Thus, Bankim bid Lo in exactly 1 round during the last 3 rounds.