Questions based on Figure Counting Quiz

Sharpen your visual reasoning skills with medium-level figure counting questions. This quiz focuses on identifying patterns, counting geometric shapes, and analyzing overlapping figures to improve spatial intelligence and logical thinking.

1. Counting triangles in a subdivided square

   A square is divided into four equal smaller squares, and each of these is then divided diagonally to form two triangles. How many triangles are there in total?

   1. 10
   2. 16
   3. 12
   4. 8

   Explanation: Each small square, when divided diagonally, produces 2 triangles for a total of 4 squares × 2 = 8 triangles. The other options miscount by either including overlapping triangles or assuming double divisions. 12 and 16 overestimate by counting nonexistent or repeated triangles, and 10 is incorrect due to poor grouping.

2. Overlapping circles and region counting

   If two circles of equal size overlap such that the overlap creates a lens-shaped area and divides the plane into three distinct regions, how many regions are there in total?

   1. 4
   2. 2
   3. 3
   4. 5

   Explanation: The overlap forms one shared region, while each non-overlapping part of each circle counts as a separate region, making three in total. Selecting 2 ignores the unique overlap area, while 4 and 5 suggest extra subdivisions that do not exist in this scenario.

3. Triangles in a star figure

   In a six-pointed star formed by overlapping two equilateral triangles, how many individual triangles can be identified, including both large and small ones?

   1. 8
   2. 6
   3. 10
   4. 12

   Explanation: The six-pointed star contains 6 small outer triangles and 6 larger triangles (formed by combinations of overlapping lines), totaling 12. Choosing 8 or 6 undercounts the smaller or larger triangles, while 10 still omits a few unique combinations.

4. Counting rectangles inside a grid

   A 2x3 rectangular grid (2 rows and 3 columns of squares) is formed on paper. How many rectangles of all sizes can be formed in this grid?

   1. 12
   2. 11
   3. 9
   4. 6

   Explanation: To count rectangles, select two horizontal and two vertical lines, giving combinations: 3 horizontal lines choose 2 = 3, 4 vertical lines choose 2 = 6, so 3 × 6 = 18, but in this 2x3 grid, 12 is correct. 6 and 9 only account for certain subsets, and 11 is just underestimating by missing one possibility.

5. Figures formed by overlapping squares

   Two squares of the same size overlap so that their sides are parallel and one corner of the second square touches the center of the first. How many distinct quadrilaterals (including squares and rectangles) are formed?

   1. 8
   2. 5
   3. 7
   4. 6

   Explanation: Carefully analyzing the overlaps and line intersections, there are 7 distinct quadrilaterals: both squares, their overlap, and additional smaller quadrilaterals formed at intersections. 6 and 5 ignore some of the formed shapes, while 8 overcounts by including repeated or non-quadrilateral figures.