Twenty five coloured beads are to be arranged in a grid comprising of five rows and five columns. Each cell in the grid must contain exactly one bead. Each bead is coloured either Red, Blue or Green.
While arranging the beads along any of the five rows or along any of the five columns, the rules given below are to be followed:
(1) Two adjacent beads along the same row or column are always of different colours.
(2) There is at least one Green bead between any two Blue beads along the same row or column.
(3) There is at least one Blue and at least one Green bead between any two Red beads along the same row or column.
Every unique, complete arrangement of twenty five beads is called a configuration.
Two Red beads have been placed in ‘second row, third column’ and ‘third row, second column’.
How many more Red beads can be placed so as to maximise the number of Red beads used in the configuration?
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Question

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.

Answer 1

To address the problem of arranging colored beads according to specified rules while maximizing the count of Red beads, a systematic approach is required:
(1) Rule 1 dictates that adjacent beads must differ in color, imposing a checkerboard constraint on both rows and columns.
(2) Rule 2 mandates at least one Green bead between any two Blue beads, maintaining an alternation requirement.
(3) Rule 3 requires at least one Blue bead and one Green bead between any two Red beads, increasing color diversity.
The initial bead placements are Red at 'second row, third column' and 'third row, second column'. The following represents one optimal configuration for placing additional Red beads within these constraints:

G	B	R	B	G
B	R	G	R	B
R	G	G	G	B
B	R	B	R	G
G	B	G	B	R

In this configuration:

Both initial Red beads satisfy all rules. Additional Red beads are strategically placed at positions (2,4), (4,2), (4,4), and (5,5) to maximize the total count while adhering to constraints.
The entire grid remains compliant with all rules, ensuring Red beads alternate with both Blue and Green beads as required.
The total count of Red beads is 8: the 2 initial beads plus 6 additional Red beads placed according to the rules and optimally.

Consequently, the maximum number of additional Red beads that can be placed is 6.

The Correct Option is A

Solution and Explanation

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