Spatial ability for AFCAT: painted cubes, dice, nets, rotation and views

Spatial ability is the part of the AFCAT reasoning section most closely tied to flying. A pilot constantly relates what they see from the cockpit to a mental picture of the aircraft's position and attitude. In the exam this becomes painted cubes, dice, cube nets, rotated figures and views of solids.

The good news is that each type has a method that does not depend on "just seeing it". Formulas handle painted cubes. Two rules handle dice. A single observation handles most nets. Grid thinking handles rotation. Candidates who say they have no spatial sense usually improve fastest here, because they had been guessing instead of using the methods.

Painted cubes

A large cube is painted on all six faces and then cut into n × n × n equal small cubes. How many small cubes have 3, 2, 1 or 0 painted faces?

The (n − 2) appears because each edge or face loses the two corner or edge layers at its ends.

Check: the four counts must add to n³.

Different colours on opposite faces. Suppose the top and bottom are red, the front and back are blue, and the left and right are green.

• Every corner cube touches one face from each pair, so the 8 corner cubes carry all three colours.
• Cubes with exactly red and blue lie on the 4 edges where a red face meets a blue face: 4 × (n − 2).

Dice

Standard dice

On a standard die, opposite faces add up to 7: 1 is opposite 6, 2 is opposite 5, and 3 is opposite 4. So 1, 2 and 3 can meet at one corner, because no two of them are opposite.

Many questions use dice that are not standard, with numbers, letters or symbols placed freely. Then you have to work out the opposite faces from the views given.

Two rules for views

1. Faces seen together are never opposite. Any two faces visible in one view are adjacent.
2. Two common faces: if two views share two faces, the third faces in those views are opposite each other.

The second rule works because the only faces touching both of two adjacent faces are the two faces that are opposite each other.

Example 1. A die shows three views: (1, 2, 3), (1, 3, 5) and (3, 4, 5). What is opposite 1?

• Views 1 and 2 share 1 and 3. So 2 and 5 are opposite.
• 3 is seen with 1, 2, 5 and 4, so 3 is adjacent to all four. That leaves 6 opposite 3.
• The only faces left are 1 and 4. 1 is opposite 4.

Pairs: 1–4, 2–5, 3–6. No other way of pairing the six faces fits all three views.

Cube nets

A net is a flat pattern of six squares that folds into a cube. Two observations settle most questions.

• In a straight line of three squares, the two end squares are opposite.
• In a straight run of four squares, squares 1 and 3 are opposite, and so are 2 and 4.

Example 2. A cross-shaped net, written row by row. A dash is an empty space.

• Column 2 is a straight run of four: 1, 3, 5, 6. So 1 is opposite 5, and 3 is opposite 6.
• Row 2 is a line of three: 2, 3, 4. So 2 is opposite 4.
• Pairs: 1–5, 2–4, 3–6.

Rotating figures

For rotation questions, place the figure on a 3×3 grid, cells numbered 1 to 9 like a phone keypad (1, 2, 3 across the top). A 90° clockwise turn sends the top row to the right column:

Example 3. An L-shape fills cells 1, 4, 7 and 8: a vertical bar down the left side with a foot pointing right at the bottom.

• Turned 90° clockwise, it fills cells 3, 2, 1 and 4, which is cells 1, 2, 3 and 4. That is a bar along the top with a short piece hanging down on the left.
• Turned 180°, it fills cells 2, 3, 6 and 9. That is a bar down the right side with a short piece pointing left at the top.
• Its mirror image fills cells 3, 6, 8 and 9. That is a J-shape, and no rotation of the L can produce it.

Views of stacked cubes

A stack of cubes is often described by a top-view grid of heights: the number in each cell is how many cubes stand on that square.

• Total cubes: add every number. 1 + 2 + 1 + 3 + 1 + 2 = 10.
• Front view: for each column, the tallest stack. Left 3, middle 2, right 2.
• Side view (from the right): for each row, the tallest stack. Front row 3, back row 2.
• Top view: every occupied square, here a 2 × 3 rectangle of six squares.

Simple solids follow the same idea. A cylinder standing on its base looks like a circle from the top and a rectangle from the front. A cone standing on its base looks like a circle with a point at its centre from the top, and a triangle from the front.

Practice set

1. A painted cube is cut into 5 × 5 × 5 small cubes. How many have no painted face?
2. In the same cube, how many have exactly two painted faces?
3. A painted cube is cut 3 times along each direction. How many small cubes have exactly one painted face?
4. A cube has red on the top and bottom, blue on the front and back, and green on the left and right. It is cut into 4 × 4 × 4 small cubes. How many small cubes have red and blue but no green?
5. A die shows three views: (2, 3, 4), (2, 4, 6) and (1, 2, 6). What is opposite 2?
6. A net is written row by row, with a dash for an empty space. Row 1: P, -, -, -. Row 2: Q, R, S, T. Row 3: -, -, U, -. Which face is opposite U?
7. In the stack from the views section, one cube is added on top of the middle stack of the back row. What does the front view show now?

Answers:

1. (5 − 2)³ = 27.
2. 12 × (5 − 2) = 36.
3. Three cuts make 4 slices, so n = 4. 6 × (4 − 2)² = 24.
4. 8. Four edges join a red face to a blue face, and each has 4 − 2 = 2 non-corner cubes.
5. 5. Views 1 and 2 share 2 and 4, so 3 is opposite 6. Face 2 is seen with 3, 4, 6 and 1, so it is opposite the only face left, 5. That leaves 1 opposite 4.
6. P. Row 2 is a run of four, so Q is opposite S and R is opposite T. P and U are the two faces left, so they are opposite each other.
7. Left 3, middle 3, right 2. The middle column's tallest stack becomes 3.

What to do next

• Write the painted-cube table for n = 6 from the formulas, and check that the total is 216.
• Take a real die, note three views of it, and practise finding opposite faces using only the two rules.
• Cut out a cross-shaped net from paper, number it, and fold it to see the straight-line rule in action.
• Revise paper folding and figure analogy, then use the AFCAT reasoning plan to schedule a mixed spatial set each week.