In this chapter, we will discuss various types of figure counting problems and their solutions, covering strategies such as identifying patterns, using symmetry, breaking down complex figures, and using formulas. Examples and practice problems will also be provided to improve your analytical reasoning skills and speed and accuracy.
Multiple Choice Questions
Que 1: What is the number of rectangles with dimensions 2×3 in the given figure?
(a) 24
(b) 38
(c) 44
(d) 52
Solution: First, we will calculate the total number of red rectangles.
Possible movement⇒ 5 positions horizontal and 4 vertical.
Total rectangles= 5×4=20
Calculate the total number of blue rectangles.
Possible movement⇒ 6 positions horizontal and 3 vertical.
Total rectangles= 6×3=18
Final count= 20+18=38
Option (b) is correct.
Que 2: What is the number of triangles in the given figure?
(a) 60
(b) 52
(c) 74
(d) 92
Solution: First, count the total number of triangles in the main figure.
Number of levels= 6
Use this series⇒ 1, 1+2, 3+3, 6+4, 10+5, 15+6, 21+7, …..
⇒ 1, 3, 6, 10, 15, 21, 28, …..
Add first six terms= 1+3+6+10+15+21= 56 …….(i)
1, 3, 6, 10, 15, 21⇐ Start from here (Color the number present in each even place)
Add these numbers= 1+6+15= 22 ……(ii)
Total triangles in the main figure= 56+22=78
Since two lines are missing in the problem figure
Hence, the 4 triangles associated with them are also missing.
The final count of the triangles= 78-4=74
Option (c) is correct.
