Spatial Aptitude

North is at the top of the map; 1st Main Road separates west from east and 5th Cross Road separates north from south.

• (A) False: the library is level with the canteen, so it is to the west of it, not the northwest.
• (B) False: the hospital is directly north of the chemistry lab.
• (C) True: the physics lab is in the north-west block and the chemistry lab is in the south-east block, so the chemistry lab is to its southeast.
• (D) False: the classrooms and the canteen are in diagonally opposite blocks.

Answer: C

The two folds run along the two diagonals, so (IV) has four layers: its top side is the four outer edges of the sheet, its left side lies on the first fold (top-left to bottom-right diagonal) and its right side on the second fold. Unfolding mirrors every cut across both diagonals.

• The top triangle appears at the middle of all four edges, each pointing towards its edge, which rules out (C).
• The half-square on the first fold opens into a full square with sides parallel to the edges, near the top-left and near the bottom-right corner, which rules out (B) (diamonds) and (D) (one square at the centre).
• The bar touching the second fold joins its mirror image to form the step shapes near the other two corners.

Answer: A

Fold the net with the open triangle at the front. The filled circle above it becomes the top, the open circle below it the bottom, and the blank squares on either side the left and right faces. The filled triangle, two squares to the right, lands at the back.

So the opposite pairs are: open triangle and filled triangle, filled circle and open circle, and the two blank faces.

• B and D show the filled triangle next to the open triangle, but those faces are opposite.
• C shows two blank faces sharing an edge, but the blank faces are opposite.
• A shows the open triangle with the filled circle above it and a blank face beside it, exactly as folded.

Answer: A

Each fold halves the sheet’s current long side, because the fold line is perpendicular to the long edge of the sheet as it is at that moment:

• Fold 1: 20 × 8 becomes 10 × 8.
• Fold 2: the long edge is now 10 cm, so 10 × 8 becomes 5 × 8.
• Fold 3: the long edge is now 8 cm, so 5 × 8 becomes 5 × 4.

The final 5 cm × 4 cm sheet has perimeter \(2(5 + 4) = 18\) cm. Halving the 20 cm side all three times would give 2.5 × 8 and a perimeter of 21 cm (option D), but by the third fold that side is no longer the long edge.

Answer: A

For AB to be a line of symmetry, every square needs a mirror image on the other side of AB, at the same distance from the line.

• Left group: the square above the line needs its image in the empty cell directly below it; the square just below the line on the left needs an image just above the line; the square two rows down needs an image two rows up. That is 3 squares.
• Right group: all three squares lie below the line, so each needs an image above it: 3 more squares.

None of these six cells is already filled, so the least number is \(3 + 3 = 6\).

Answer: A

Three cuts are enough: cut through the centre of the cube once parallel to each pair of opposite faces. Each cut halves every piece it passes through, giving \(2 \times 2 \times 2 = 8\) identical small cubes.

Fewer cuts cannot work. A straight cut splits each existing piece into at most two, so one cut gives at most 2 pieces and two cuts at most 4. At least 3 cuts are therefore needed, and 4, 7 and 8 are all more than the minimum.

Answer: A