Question 1

A cube has faces numbered 1 to 6. Face 1 is opposite to face 6, and face 2 is opposite to face 5. When face 3 is on top, which face is at the bottom?

Accepted Answer

This question tests understanding of opposite faces on a cube using given constraints.

Question 2

A figure consists of a 3×3 grid of squares (9 small squares total). How many squares of all possible sizes can be counted in this grid?

Accepted Answer

This question tests the ability to count squares of different sizes in a grid, recognizing that larger squares can be formed by combining smaller ones.

Question 3

A standard die shows opposite faces summing to 7. If face showing 2 is facing you, and face showing 3 is on top, which face is facing away from you?

Accepted Answer

This tests knowledge of standard die configuration and spatial orientation.

Question 4

How many squares of all sizes are present in a 4×4 grid?

[A 4×4 grid of small squares is shown]

Accepted Answer

This question tests the ability to count squares of different sizes within a subdivided rectangular grid.

Question 5

In a complex figure, there is a large square divided into 4 equal smaller squares. Each smaller square is further divided into 4 equal triangles by drawing both diagonals. Additionally, a circle is inscribed in the top-left smaller square. How many total triangles are present in the entire figure?

Accepted Answer

This question tests the ability to systematically count overlapping geometric shapes without missing or double-counting. Students must identify all triangles formed by the diagonals while accounting for composite triangles.

Question 6

A square is divided into 9 equal smaller squares (3×3 grid). A diagonal is drawn in each of the 4 corner squares, and a circle is inscribed in the center square. How many triangles are formed in the entire figure?

Accepted Answer

This question tests the ability to count triangles in a partially subdivided grid, distinguishing between triangles formed by diagonals and those that don't exist (e.g., those that would overlap with the circle).

Question 7

A rectangular grid contains 5 rows and 6 columns of small squares. How many rectangles of all possible sizes can be formed in this grid?

Accepted Answer

This question tests understanding of combinatorial counting in grid-based figures. Students must recognize that rectangles are formed by selecting 2 horizontal lines from available lines and 2 vertical lines from available lines.

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