Logical Reasoning – Twenty five coloured beads are to be arranged

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.

Q.1 The total number of possible configurations using beads of only two colours is:?

Answer: 2

Q.2 What is the maximum possible number of Red beads that can appear in any configuration?

Answer: 9

Q.3 What is the minimum number of Blue beads in any configuration?

Answer: 6

Q.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?

Answer: 6

Solutions:
from point 3 it is clear that if we have to use only two colors then it should be blue and green only . possible arrangements are as given below:

B	G	B	G	B
G	B	G	B	G
B	G	B	G	B
G	B	G	G	G
B	G	B	G	B

OR

G	B	G	B	G
B	G	B	G	B
G	B	G	B	G
B	G	B	G	B
G	B	G	B	G

ANS:2

Q.2 To maximize the number of Red beads in any configuration we need to use minimum blue or green beads between them the possible arrangement will be as below -:

R	B	G	R	B
G	R	B	G	R
B	G	R	B	G
R	B	G	R	B
G	R	B	G	R

Maximum possible red beads = 9
ANS:9

Q.3 To minimize the number of blue beads we need to maximize number of green beads the arrangement will be as below :

R	G	B	G	R
G	R	G	B	G
B	G	R	G	B
R	B	G	R	G
G	R	B	G	R

Minimum Blue beads = 6
ANS:6

Q.4 Lets place two Red beads have been placed in ‘second row, third column’ and ‘third row, second column’

		R
	R

Now lets try to place maximum number of red beads as below (we know minimum 2 places are need to be filled by other colors one blue and one green )

R			R
		R
	R			R
R			R
		R

Now let’s fill the other box with other color

R	G	B	R	G
G	B	R	G	B
B	R	G	B	R
R	G	B	R	G
G	B	R	G	B

Maximum red beads = 8
New red beads = 6
ANS:6

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