RAS question
How many total squares can be counted in a 4×4 grid (a square divided into 16 equal smaller squares)?
Correct answer: (A) 30.
A 4x4 grid contains 30 axis-parallel squares in total.
Explanation
In a square grid, the count is not limited to the 16 smallest unit squares. Wolfram MathWorld states that the number of axis-parallel squares contained in an n x n grid is the square pyramidal number, which is the sum of the squares from 1 to n. For n = 4, that means counting every possible size: sixteen 1x1 squares, nine 2x2 squares, four 3x3 squares and one 4x4 square. Adding these gives 16 + 9 + 4 + 1 = 30, so option A is the only count that includes all square sizes in the grid.
Why the other options are wrong
- (B) Option B counts only the sixteen 1x1 unit squares and leaves out the larger 2x2, 3x3 and 4x4 squares.
- (C) Option C effectively stops at 16 + 9, so it misses the four 3x3 squares and the single 4x4 square.
- (D) Option D can arise from adding 16 and 4, but that skips the nine 2x2 squares and the whole 4x4 square.
Concept
This tests systematic counting under Reasoning & Mental Ability: the visible unit squares are only the first layer, and the same n x n grid formula forces the candidate to count by size. It recurs in RAS-style reasoning because a quick visual count is tempting but incomplete.