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 maximum possible number of Red beads that can appear in any configuration? [This Question was asked as TITA]
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 maximum possible number of Red beads that can appear in any configuration? [This Question was asked as TITA]
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 maximum possible number of Red beads that can appear in any configuration? [This Question was asked as TITA]
Updated On: Jul 3, 2024
- 5
- 7
- 9
- 11
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The Correct Option is C
Approach Solution - 1
Between Any two Red beads there must be at least two Beads. So any Row or column there can be maximum two red beads. If we place two red beads in each row then two columns will have three red bead which cannot be accepted.
The above configuration is not correct.
So in the third row we will place only one Red bead at the middle of the third row. Also we will adjust other rows so that between any two Red beads there are at least two beads in any column.
So maximum 9 Red beads are possible in any configuration. At remaining places Green and Blue coloured beads can be placed in such way that all the conditions given are satisfied. There are multiple configurations are possible. One of the configurations is given as below.
So, the correct option is (C): 9.
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Approach Solution -2
To maximize the number of red beads, we need to minimize the number of blue and green beads while still satisfying the given conditions.
One optimal configuration is as follows:
R G R G R
G R G R G
R G R G R
G R G R G
R G R G R
In this configuration, there are:
- 9 red beads,
- 8 green beads,
- 8 blue beads.
Therefore, the maximum possible number of red beads in any configuration is 9
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