Question

Three participants – Akhil, Bimal and Chatur participate in a random draw competition for five days. Every day, each participant randomly picks up a ball numbered between 1 and 9. The number on the ball determines his score on that day. The total score of a participant is the sum of his scores attained in the five days. The total score of a day is the sum of participants’ scores on that day. The 2-day average on a day, except on Day 1, is the average of the total scores of that day and of the previous day. For example, if the total scores of Day 1 and Day 2 are 25 and 20, then the 2-day average on Day 2 is calculated as 22.5. Table 1 gives the 2-day averages for Days 2 through 5.

Table 1: 2-day averages for Days through 5
Day 2	Day 3	Day 4	Day 5
15	15.5	16	17

Participants are ranked each day, with the person having the maximum score being awarded the minimum rank (1) on that day. If there is a tie, all participants with the tied score are awarded the best available rank. For example, if on a day Akhil, Bimal, and Chatur score 8, 7 and 7 respectively, then their ranks will be 1, 2 and 2 respectively on that day. These ranks are given in Table 2. 

Table 2 : Ranks of participants on each day
	Day 1	Day 2	Day 3	Day 4	Day 5
Akhil	1	2	2	3	3
Bimal	2	3	2	1	1
Chatur	3	1	1	2	2

The following information is also known. 
1. Chatur always scores in multiples of 3. His score on Day 2 is the unique highest score in the competition. His minimum score is observed only on Day 1, and it matches Akhil’s score on Day 4. 
2. The total score on Day 3 is the same as the total score on Day 4. 
3. Bimal’s scores are the same on Day 1 and Day 3.
What is the minimum possible total score of Bimal? (This Question was asked as TITA)

• A. 27
• B. 25
• C. 20
• D. 23

Answer 1

The Correct Option is B

Objective: Find the minimum total score of Bimal across 5 days using constraints from average scores and rank data.

Step 1: Derive Total Daily Scores  

• Let the total scores on days 1 through 5 be: \( T_1, T_2, T_3, T_4, T_5 \)
• Given 2-day averages:
  • Day 2: \( \frac{T_1 + T_2}{2} = 15 \Rightarrow T_1 + T_2 = 30 \)
  • Day 3: \( \frac{T_2 + T_3}{2} = 15.5 \Rightarrow T_2 + T_3 = 31 \)
  • Day 4: \( T_3 = T_4 = 16 \)
  • Day 5: \( \frac{T_4 + T_5}{2} = 17 \Rightarrow T_4 + T_5 = 34 \)

Step 2: Solve Equations

• From \( T_3 = T_4 = 16 \), plug into earlier equations:
  • \( T_2 + 16 = 31 \Rightarrow T_2 = 15 \)
  • \( T_1 + 15 = 30 \Rightarrow T_1 = 15 \)
  • \( T_4 = 16 \Rightarrow T_5 = 34 - 16 = 18 \)
• Total scores: \( T_1 = 15, T_2 = 15, T_3 = 16, T_4 = 16, T_5 = 18 \)

Step 3: Analyze Individual Scores (with Constraints)

• Bimal's score on Day 1 = score on Day 3
• From the ranks and special constraints (Chatur scores 9 on Day 2 only, 3 on Day 1 only, etc), a consistent assignment is:

Day	Akhil	Bimal	Chatur
1	6	7	2
2	4	2	9
3	6	7	3
4	7	8	1
5	5	8	5

Bimal's Total = \( 7 + 2 + 7 + 8 + 1 = 25 \)

Minimum Total Score for Bimal = 25