Question

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 maximum possible number of Red beads that can appear in any configuration? [This Question was asked as TITA]

• A. 5
• B. 7
• C. 9
• D. 11

Answer 1

The Correct Option is C

The question requires us to arrange 25 beads (Red, Blue, or Green) in a 5x5 grid, adhering to specific rules about bead placement in rows and columns. We aim to maximize the number of Red beads. Let's proceed methodically: 

Each row and column contains 5 beads, all different in color initially, complying with Rule 1.

Assume the worst-case scenario for the given rules where we can have the maximum number of Red beads:

To satisfy Rule 1, Red beads must alternate with other colors, as no two adjacent beads can be the same.

Rule 2 ensures that no Blue beads are adjacent without at least one Green bead in between, thus Blue and Green alternate when possible.

Rule 3, for Red beads, necessitates at least one Blue and one Green bead between them, dictating spacing for additional Red beads in each row/column.

Given these rules, if we aim to have the maximum number of Red beads, the optimal arrangement has alternate rows starting with Red and alternating with placements between Blue and Green. The configuration can be optimized as follows:

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 keeps a minimum of Blue and Green beads between the Red ones, thus maximizing Reds within the given constraints. By counting the Red beads in this pattern, we find there are 9 Red beads, satisfying all the rules:

1. Each placement respects Rule 1, with Red never adjacent to another Red.
2. Blue is never adjacent without Green in between, satisfying Rule 2.
3. Red is never adjacent without at least one Blue and one Green in between, satisfying Rule 3.

Therefore, with careful arrangement and rule consideration, the maximum number of Red beads in any configuration is 9.

Answer 2

To maximize the number of red beads, we need to minimize the number of blue and green beads while still satisfying the given conditions.

One optimal configuration is as follows:
R G R G R
G R G R G
R G R G R
G R G R G
R G R G R
In this configuration, there are:

• 9 red beads,
• 8 green beads,
• 8 blue beads.

Therefore, the maximum possible number of red beads in any configuration is 9