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 minimum number of Blue beads in any configuration?
[This Question was asked as TITA]

• A. 2
• B. 5
• C. 3
• D. 6

Answer 1

The Correct Option is D

To solve the problem of determining the minimum number of Blue beads in any configuration, we must adhere to the given rules while arranging the beads. We have a 5x5 grid resulting in 25 cells. Each bead must be Red, Blue, or Green. The constraints are:

1. Two adjacent beads must have different colors. 
2. There must be at least one Green bead between any two Blue beads.
3. There must be at least one Blue and one Green bead between any two Red beads.

To minimize the number of Blue beads, we must consider the maximum use of Red and Green beads under these constraints. Analyzing the scenario:

• When using Red beads, each Red must be separated by at least two other beads (one Blue and one Green). Hence, the sequence around any Red can be R-G-B or R-B-G.
• For Blue beads, every Blue must have at least one Green bead between another Blue, requiring sequences like B-G.

Given the structure of a row or column:

1. Start with a Red bead, followed by a Green and a Blue as necessary:
   • R-G-B-G-R or similar.
2. By introducing conditions, we can create sequences maximizing the use of non-Blue beads, spacing them according to constraints, e.g.,
   • G-R-B-G | R-G-B-G | G-R | B-G | R...

Applying the constraints:

• Using these optimal arrangements, several attempts reveal:
• Each row and column must include enough Blue beads, as their placement determines separating conditions for Red and Green maximizing compliance with provided rules.

After trying logical patterns and minimal configurations ensuring these constraints are always followed, using at least six Blue beads is essential to satisfy all constraints across a 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 strategic placement of Blue beads, ensures conditions are met minimally, thereby demanding at least 6 Blue beads. Hence, the minimum number of Blue beads in any configuration is 6.

Answer 2

Given a maximum of 9 red beads, we aim to fill the remaining space with green and blue beads while minimizing the number of blue beads used.


Hence number of blue beads is 6.