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Lesson 3 of 3 · Count triangles / squares / rectangles

Rectangles

Non-verbal Reasoning · for IPMAT Indore, IPMAT Rohtak and JIPMAT

The exam method

  1. Grids. squares size by size (4×44 \times 4: 16+9+4+1=3016 + 9 + 4 + 1 = 30); rectangles (m+12)(n+12)\binom{m+1}{2}\binom{n+1}{2}.

Worked examples

Example 3Easy

Rectangles in a 4 by 2 grid.

Formulas you need

rectangles in an m×n grid=(m+12)(n+12)\text{rectangles in an } m \times n \text{ grid} = \binom{m+1}{2}\binom{n+1}{2}

squares in an n×n grid=12+22+⋯+n2\text{squares in an } n \times n \text{ grid} = 1^2 + 2^2 + \dots + n^2

squares in an m×n grid (m≥n)=∑k=1n(m−k+1)(n−k+1)\text{squares in an } m \times n \text{ grid } (m \ge n) = \sum_{k=1}^{n} (m-k+1)(n-k+1)

Picked from the Count triangles / squares / rectangles note for this micro-concept automatically. The whole note: Count triangles / squares / rectangles.