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: Two adjacent beads along the same row or column are always of different colours. There is at least one Green bead between any two Blue beads along the same row or column. 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?

Logical Reasoning

Grids

Hard

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:

Two adjacent beads along the same row or column are always of different colours.

There is at least one Green bead between any two Blue beads along the same row or column.

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?

Entered answer:

✅ Correct Answer: 9

Slide 1/2

Understanding the set :-

It says, there are 25 beads to be arranged in a grid comprising of five rows and five columns.

Each bead is coloured either Red, Blue or Green.

we need to arrange the beads according to the rules given :

a) Two adjacent beads are always of different colour

b) There is at least one Green bead between any two Blue beads

c) There is at least one Blue and at least one Green bead between any two Red beads

This set gives information about 25 beads, which we need to arrange in a grid of 5 rows and 5 columns,

so there will be n different configurations we can make of these 25 beads arrangement.

Therefore, this is a question based set,

which we need to solve according to the question's demand.

Question ask us to maximize the number of red beads -

Now, we know 1 Red needs to have at least 1 blue and 1 green between them, to maximize, we will need to have (ideally) exactly 1 blue and 1 green. (Clue 3)

therefore, we will try for red beads to not come in contact in adjacent cell, to avoid use of blue and green beads.

thus, Move along the diagonal as you are trying to fill the Red beads since that way, we can also ensure that 2Rs do not come in contact in adjacent cells

Red			Red
	Red			Red
		Red
Red			Red
	Red			Red

Let's see if this is feasible with the given constraints. We will now need to fill in the G and B to see if it is feasible.

Red	Green	Blue	Red	Green
Green	Red	Green	Blue	Red
Blue	Green	Red	Green	Blue
Red	Blue	Green	Red	Green
Green	Red	Blue	Green	Red

It is indeed feasible,

by now you would have already noticed a pattern that moving along the diagonal (especially because of the non-repeating colours condition) is absolutely crucial to understanding the pattern in the grid.

Here as well, you can see that the Blues and Greens are along the diagonal.

Therefore, we can see that there are 9 Red colored beads in the above arrangement.

Question 1

The total number of possible configurations using beads of only two colors is:

Question 3

What is the minimum number of Blue beads in any configuration?

Question 4

Two Red beads have been placed in 'second row, third column' and 'third row, second column'. How many more Red beads can be placed so as to maximise the number of Red beads used in the configuration?

What is the maximum possible number of Red beads that can appear in any configuration?

Solution

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