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.
What is the minimum number of Blue beads in any 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 determine the minimum count of Blue beads in any valid arrangement on a 5x5 grid (25 cells), the following rules must be applied. Each cell contains a Red, Blue, or Green bead.

Adjacent beads must differ in color.
A Green bead must separate any two Blue beads.
Both a Blue and a Green bead must separate any two Red beads.

Minimizing Blue beads requires maximizing Red and Green beads within these constraints. Analysis shows:

Red beads necessitate at least two beads (one Blue, one Green) between them, forming patterns like R-G-B or R-B-G.
Blue beads require separation by at least one Green bead, in patterns like B-G.

Considering a row or column structure:

Starting with Red, subsequent beads are placed to fulfill requirements:
- Example: R-G-B-G-R.
Sequences can be optimized to maximize non-Blue beads, adhering to spacing rules, such as:
- G-R-B-G | R-G-B-G | G-R | B-G | R...

Applying these rules:

Optimal arrangements indicate that the placement of Blue beads is critical for satisfying Red and Green separation rules.
Sufficient Blue beads are necessary in each row and column to meet these separation requirements.

Through logical pattern construction and minimal compliant configurations, it is determined that at least six Blue beads are required to satisfy all constraints across the grid.

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

This configuration, with strategically placed Blue beads, meets all conditions minimally, establishing the requirement of at least 6 Blue beads. Therefore, the minimum number of Blue beads in any valid configuration is 6.

The Correct Option is D

Solution and Explanation

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