Mensa IQ Test Spatial Reasoning 3 — Questions and Answers

Question 1: How many triangles are in a figure made of a large triangle divided into 4 equal smaller triangles?

4
5 (Correct answer)
6
7

Correct answer: 5

There are 4 small triangles plus the 1 large outer triangle = 5 triangles total.

A triangle divided into 4 equal parts contains: 4 small triangles (the subdivisions) + 1 large triangle (the whole shape) = 5 triangles total. Always look for composite shapes in counting problems.

Question 2: If a paper is folded in half twice and then a hole is punched through all layers, how many holes appear when unfolded?

2
4 (Correct answer)
6
8

Correct answer: 4

Folding in half twice creates 4 layers. One punch through all 4 layers creates 4 holes when unfolded.

First fold: 2 layers. Second fold: 4 layers. Punching once through 4 layers creates 4 holes. When unfolded, the holes appear symmetrically — 4 holes arranged in a 2×2 pattern around the center of the paper.

Question 3: A solid cube is painted red on all sides and cut into 27 equal smaller cubes. How many smaller cubes have exactly 2 painted faces?

6
8
12 (Correct answer)
18

Correct answer: 12

Edge cubes (not corners) have exactly 2 painted faces. A 3×3×3 cube has 12 edges, each contributing 1 such cube = 12.

In a 3×3×3 cube: Corner cubes (3 painted faces) = 8. Edge cubes excluding corners (2 painted faces) = 12 edges × 1 cube each = 12. Face-center cubes (1 painted face) = 6. Center cube (0 faces) = 1. Total = 8+12+6+1 = 27. The answer is 12.

Question 4: What is the minimum number of straight cuts needed to divide a circle into 8 equal pieces?

3
4 (Correct answer)
7
8

Correct answer: 4

Each straight cut through the center adds 2 equal sectors. 1 cut = 2 pieces, 2 cuts = 4 pieces, 3 cuts = 6 pieces, 4 cuts = 8 pieces.

Cuts through the center create equal sectors. 1 cut = 2 pieces, 2 cuts = 4, 3 cuts = 6, 4 cuts = 8. To divide a circle into 8 equal pieces with straight cuts through the center, you need 4 cuts (each 45° apart).

Question 5: Which shape has exactly 4-fold rotational symmetry?

Equilateral triangle
Regular pentagon
Square (Correct answer)
Regular hexagon

Correct answer: Square

A square looks identical after rotations of 90°, 180°, 270°, and 360° — that is 4-fold rotational symmetry.

4-fold rotational symmetry means the shape looks the same at 0°, 90°, 180°, 270°. A square satisfies this. Equilateral triangle has 3-fold (120°), regular pentagon has 5-fold (72°), regular hexagon has 6-fold (60°).

Question 6: A cube has a space diagonal from one corner to the opposite corner. If the cube has side length 2, what is the length of this space diagonal?

2√2
2√3 (Correct answer)
4
2√5

Correct answer: 2√3

Space diagonal of a cube with side a = a√3. With a=2: 2√3.

For a cube with side length a, the space diagonal = √(a²+a²+a²) = a√3. With a = 2: space diagonal = 2√3 ≈ 3.46. This uses the 3D Pythagorean theorem applied to the cube's three dimensions.

Mensa IQ Test Spatial Reasoning 3

How many triangles are in a figure made of a large triangle divided into 4 equal smaller triangles?