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.
The total number of possible configuration using beads of only two colours is:
[This Question was asked as TITA]

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

To solve this problem, we need to explore how the two-color configurations can be produced while adhering to the given constraints. Since we are only using two colors out of the possible three (Red, Blue, Green), we will analyze each possible color combination.

1. Using Red and Green:
   • Rule 1 (adjacent beads different): This rule can be easily complied with by alternating Red and Green in each row and column.
   • Rule 2 (Green between two Blues): This rule is irrelevant since Blue is not used in this configuration.
   • Rule 3 (Blue and Green between Reds): This rule also does not apply as Blue is not used.
   • Conclusion: A consistent alternating pattern of Red and Green (e.g., a checkerboard pattern) will generate a valid configuration.
2. Using Blue and Green:
   • Rule 1: Like above, alternate Blue and Green in rows and columns.
   • Rule 2: Automatically satisfied as Green is placed next to all Blues.
   • Rule 3: Irrelevant since Red isn't included.
   • Conclusion: An alternating pattern (e.g., checkerboard) satisfies the rules.
3. Using Red and Blue:
   • Rule 1: Can be satisfied by alternating the beads.
   • Rule 2: Green is needed between Blues, but Green is unavailable.
   • Rule 3: Green is needed between Reds, but Green is unavailable.
   • Conclusion: Violates Rules 2 and 3, so no valid configuration exists.

Initially, two configurations possible: one each for Red-Green and Blue-Green. Thus, the total possible configurations using only two colors are 2.

Answer 2

We are arranging beads in a row or column with only two colors allowed per row/column.

However, there's a special restriction on red-colored beads:

Between any two red beads, there must be:

At least two beads in between

And these two beads must include at least one green and at least one blue

This is a strict spacing rule that red beads cannot be adjacent, and even cannot be placed with just one bead in between — and the beads in between must be of both green and blue colors.