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How many squares are in a 3x3 grid?
9
12
14
16
14
In an NxN grid, sum the squares of natural numbers from 1 to N: (1² + 2² +... + N²). For a 3x3 grid, calculate 1² + 2² + 3² = 1 + 4 + 9 = 14.
In an NxN grid, sum the squares of natural numbers from 1 to N: (1² + 2² +... + N²). For a 3x3 grid, calculate 1² + 2² + 3² = 1 + 4 + 9 = 14.
Identify individual small squares
A 3x3 grid consists of 9 unit squares (1x1 size).
Identify medium squares
There are 4 squares of size 2x2. These are formed by combining four 1x1 squares in a block.
Identify the largest square
There is 1 square of size 3x3, which is the entire boundary of the grid.
Calculate total
Adding these counts together: 9 (size 1x1) + 4 (size 2x2) + 1 (size 3x3) = 14 total squares.
A: 9 only counts the smallest unit squares and ignores larger 2x2 and 3x3 squares. B: 12 is an incorrect total likely resulting from missing squares or miscounting the 2x2 blocks. D: 16 is incorrect as it overestimates the total by including non-existent configurations.
C is correct because the sum of squares of integers from 1 to 3 (1² + 2² + 3²) equals 14.
For rectangular grids of size MxN, the formula is [MN] + [(M-1)(N-1)] + [(M-2)*(N-2)]... until one dimension becomes 1. This prevents manual counting errors.
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