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 minimum number of Blue beads 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 minimum number of Blue beads in any configuration?

Entered answer:

✅ Correct Answer: 6

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 says,

minimum number of Blue beads in any configuration.

Procedure to solve this question will be exactly like the previous one.

We need to include a minimum number of Bs in the grid,

meaning, ideally we should include only 1 B per row. Let's attempt to do that.

Before we begin, what seems to be clear from the grid so far is that,

if we are trying to maximize any colour, it is possible only if we move through the main diagonal.

Clearly since that is not what we are trying to achieve for Blues, we should make sure that in the initial arrangement, Blue is not present in the middle cell / along the main diagonal

But while filling only 1 colour, we always move through the diagonal of the starting point. An initial arrangement might look like this:

		Blue
			Blue
Blue				Blue
	Blue
		Blue

Now let's try to fill in colours and see what is feasible.

While doing so, we need to keep in mind that since we are minimizing Blue, it automatically means we are trying to maximize Red and Green.

Which means that along the main diagonal should be beads with either Red or Green.

So the only feasible solution is:

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

So we can see that there are 6 Blue colored beads in the above arrangement.

Question 1

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

Question 2

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

Solution

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