The objective is to arrange 25 beads (Red, Blue, or Green) in a 5x5 grid according to specific row and column placement rules, with the goal of maximizing the number of Red beads. The process is as follows:
Rule 1 states that each row and column must contain 5 beads, all initially different in color.
We consider the scenario most favorable for maximizing Red beads under the given rules.
To comply with Rule 1, Red beads must be placed such that no two adjacent beads share the same color.
Rule 2 mandates that Blue beads cannot be adjacent; at least one Green bead must separate them, implying alternation between Blue and Green where feasible.
Rule 3 requires at least one Blue bead and one Green bead to be positioned between any two Red beads in each row/column, dictating the spacing for additional Red beads.
Given these constraints, the optimal arrangement for maximizing Red beads involves alternating rows starting with Red, with placements between Blue and Green beads. The optimized configuration is:
| R | G | R | B | R |
| G | R | B | R | G |
| R | B | R | G | R |
| B | R | G | R | B |
| R | G | R | B | R |
This configuration ensures a minimum separation of Blue and Green beads between Red beads, thus maximizing the Red bead count within the stipulated limitations. Counting the Red beads in this pattern reveals a total of 9 Red beads, which satisfies all rules:
Consequently, through careful arrangement and adherence to the rules, the maximum achievable number of Red beads in any configuration is 9.
Ananya Raga, Bhaskar Tala, Charu Veena, and Devendra Sur are four musicians. Each of them started and completed their training as students under each of three Gurus- Pandit Meghnath, Ustad Samiran, and Acharya Raghunath between 2013 and 2024, including both the years. Each Guru trains any student for consecutive years only, for a span of 2, 3, or 4 years, with each Guru having a different span. During some of these years, a student may not have trained under these Gurus; however, they never trained under multiple Gurus in the same year.
In none of these years, any of these Gurus trained more than two of these students at the same time. When two students train under the same Guru at the same time, they are referred to as Gurubhai, irrespective of their gender.
The following additional facts are known.
1. Ustad Samiran never trained more than one of these students in the same year.
2. Acharya Raghunath did not train any of these students during 2015-2018, as well as during 2021-24.
3. Ananya and Devendra were never Gurubhai; neither were Bhaskar and Charu. All other pairs of musicians were Gurubhai for exactly 2 years.
4. In 2013, Ananya and Bhaskar started their trainings under Pandit Meghnath and under Ustad Samiran, respectively.