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:
Solution and Explanation
Given the constraint of using only two colors, beads in any row or column must alternate. For example:
| 1 | 2 | 1 | 2 | 1 |
This pattern prevents placing Red beads at positions 1 or 2, as there must be at least two beads (one green and one blue) between any two Red beads. Therefore, only Green and Blue colored beads can be used.
Two possible configurations exist:
Configuration 1: A Green bead at the top-left corner.
| 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 |
Configuration 2: A Blue bead at the top-left corner.
| B | G | B | G | G |
| G | B | G | B | B |
| B | G | B | G | G |
| G | B | G | B | B |
| B | G | B | G | G |
Thus, there are 2 possible configurations.
What is the maximum possible number of Red beads that can appear in any configuration?
Solution and Explanation
A constraint dictates that at least two beads must separate any two red beads. Consequently, each row and column can contain a maximum of two red beads. Placing two red beads in every row would result in three red beads in two columns, which is unacceptable.
| R | R | |||
| R | R | |||
| R | R | |||
| R | R | |||
| R | R |
The preceding arrangement is invalid. Therefore, the third row will accommodate only one red bead, positioned centrally. Other rows will be adjusted to ensure that at least two beads separate any two red beads within any column.
| R | R | |||
| R | R | |||
| R | ||||
| R | R | |||
| R | R |
This configuration allows for a maximum of 9 red beads. Green and blue beads can be placed in the remaining positions, adhering to all specified conditions. Multiple valid configurations exist. One such configuration is presented below.
| R | G | B | R | G |
| G | R | G | B | R |
| B | G | R | G | B |
| R | B | G | R | G |
| G | R | B | G | R |
The maximum number of red beads possible is 9.
What is the minimum number of Blue beads in any configuration?
Solution and Explanation
To minimize the quantity of Blue beads, we must maximize the quantity of Red and Green beads. Based on the prior solution, the maximum number of Red beads is 9. In rows containing two Red beads, we will add two Green beads and one Blue bead. In rows with only one Red bead, we will add two Green beads and two Blue beads. Consequently, the minimum number of Blue beads will be 6.
| R | G | B | R | G |
| G | R | G | B | R |
| B | G | R | G | B |
| R | B | G | R | G |
| G | R | B | G | R |
Therefore, the result is 6.
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?
Solution and Explanation
The objective is to position red beads so that no two beads are adjacent, including diagonally.
Valid Configuration:
| R | R | |||
| R | ||||
| R | R | |||
| R | R | |||
| R |
The displayed configuration demonstrates red beads (R) placed without any two being adjacent, adhering to the stipulated condition.
Answer:
\[ \boxed{6 \text{ red beads}} \] is the maximum number that can be placed on the grid according to the specified rules.
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