GATE DA 2024 | Question 2

MCQ · 1 mark · −0.33

The 15 parts of the given figure are to be painted such that no two adjacent parts with shared boundaries (excluding corners) have the same color. The minimum number of colors required is

An ellipse with a square inside it and a circle at the centre of the square. Three radii, to the top, lower left and lower right, split the circle into three sectors; four lines from the corners of the square to the circle split the band around the circle into top, right, bottom and left parts; and two lines from each corner of the square to the ellipse split the outer band into four side parts and four corner parts.
  1. Option A: 4 (correct)
  2. Option B: 3
  3. Option C: 5
  4. Option D: 6

Answer

(A)

Explanation

The three sectors of the circle all touch one another, so they need three different colours: say 1 (upper left), 2 (bottom) and 3 (upper right).

In the band around the circle, the left part touches sectors 1 and 2, so it must be 3, and the right part touches sectors 2 and 3, so it must be 1. The part below the circle touches sector 2, the left part (3) and the right part (1), so it needs a fourth colour. Three colours are not enough.

Four are: colour the top band part 2 and the bottom band part 4, the outer top and bottom parts 1, the outer left and right parts 2, and all four outer corner parts 3.

Answer: A