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 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:

1. Rule 1 is met as Red beads are never adjacent to another Red bead.
2. Rule 2 is satisfied, as Blue beads are always separated by at least one Green bead.
3. Rule 3 is upheld, as Red beads are always separated by at least one Blue and one Green bead.

Consequently, through careful arrangement and adherence to the rules, the maximum achievable number of Red beads in any configuration is 9.