CAT 2018 Slot 1 — DILR Question 18
Answer the following question based on the information given below.
You are given an n × n square matrix to be filled with numerals so that no two adjacent cells have the same numeral. Two cells are called adjacent if they touch each other horizontally, vertically or diagonally. So a cell in one of the four corners has three cells adjacent to it, and a cell in the first or last row or column which is not in the corner has five cells adjacent to it. Any other cell has eight cells adjacent to it.
What is the minimum number of different numerals needed to fill a 5 × 5 square matrix?
Answer & solution
Answer: 4
A number say ‘x’ can be filled in four corner cells. Another number say ‘y’ can be used in the central cell. Now the remaining cells are middle cells of top row, bottom row, left column and right column. Two more numerals can be used to fill these cells. Assume that ‘z’ and ‘w’ are those numerals.
Thus,
So, at least 4 different numerals are required for a 3 × 3 matrix.
Top row i.e., the first row of the 5 × 5 matrix can be (x, z, x, z, x). Similarly, other cells from the second and the third row can be filled.
For the first column i.e., the leftmost column, the fourth cell from top can be filled with the number from the second cell from top i.e. w and the bottom cell can be filled with the number from the top most cell of the column i.e., x. Similarly, remaining cells of the remaining columns can be filled.
Thus, four different numerals are sufficient to fill a 5 × 5 matrix.
Therefore, the required answer is 4.
Answer: 4