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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.
Question 2 : What is the maximum possible number of Red beads that can appear in any conguration?
The idea here is to maximize the number of Reds in each row, this question will require us to draw the grid
Step 1: Try to draw as many Rs in a row as possible, since we know that 1R needs to have at least 1B and 1G between them, to maximize, we will need to have (ideally) exactly 1B and 1G
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
Step 2: 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.
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 Bs and Gs are along the diagonal.
The question is "What is the maximum possible number of Red beads that can appear in any conguration?"
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