GATE 2024 PI – Question 2

Correct answer: (A) 4

Explanation

Three colours are not enough, and four are enough.

**Why 3 is too few.** The three sectors of the small circle all touch each other, so they need three different colours, say 1, 2 and 3. Look at the four regions between the circle and the square:

• The top region touches the two upper sectors (1 and 2), so it must be 3.
• The left region touches the upper-left sector (1) and the bottom sector (3), so it must be 2.
• The right region touches the upper-right sector (2) and the bottom sector (3), so it must be 1.
• The bottom region touches the bottom sector (3) and also the left (2) and right (1) regions, so it needs a **fourth** colour.

**Why 4 is enough.** Any map drawn on a plane can be coloured with four colours (the four colour theorem), so the outer ring can be coloured too. The minimum is 4.