"Spatial Aptitude" — Practice

Question 1

A square sheet is folded in half three times, and one hole is punched through the folded stack away from every crease. How many holes appear when the sheet is unfolded?

Accepted Answer

C

Question 2

Which of these words appears unchanged in a mirror, assuming each letter is drawn symmetrically?

Accepted Answer

A

Question 3

A cube of side 3 units is painted on all faces and cut into 27 unit cubes. How many have no painted face?

Accepted Answer

B

Question 4

A cube net consists of a horizontal row of four squares labelled A, B, C, D from left to right, with square E attached above B and square F attached below B. Which face is opposite C?

Accepted Answer

Use the rule that faces separated by exactly one face in a straight line are opposite.

In the row A, B, C, D, the pairs separated by one face are A and C, and B and D.

So A is opposite C, and B is opposite D.

That leaves E and F, which are separated by exactly one face — namely B — in the vertical line E, B, F. So E is opposite F.

The face opposite C is A.

The check worth doing is that the three pairs use all six faces exactly once, which they do: A-C, B-D, E-F. If a proposed pairing leaves a face in two pairs or in none, the reading of the net was wrong.

Question 5

How many squares of any size are there in a 4 by 4 grid of unit cells?

Accepted Answer

Count by size class, from the largest down.

Squares of side 4: only one position fits, so 1.

Squares of side 3: the top-left corner can sit in a 2 by 2 region of positions, so 4.

Squares of side 2: the corner can sit in a 3 by 3 region, so 9.

Squares of side 1: these are the unit cells themselves, so 16.

Total: 1 + 4 + 9 + 16 = 30.

The pattern generalises: an n by n grid contains the sum of squares from 1 to n. For n = 4 that is 30, and for n = 3 it is 14, which is why a 3 by 3 grid has 14 squares among its 36 rectangles.

Question 6

A square tile carries a small black dot in its top-left corner and a diagonal line from the top-right corner to the bottom-left corner. The tile is rotated 270 degrees clockwise. Where is the dot, and what is the orientation of the line?

Accepted Answer

Track the dot first, since it is the asymmetric feature.

Under a 90-degree clockwise rotation, top-left moves to top-right. Under 180 degrees it moves to bottom-right. Under 270 degrees clockwise it moves to bottom-left.

So the dot ends in the bottom-left corner.

Now the line. It runs from top-right to bottom-left, which is one of the two diagonals. A 90-degree rotation maps one diagonal onto the other, and a 180-degree rotation maps each diagonal onto itself.

270 degrees is an odd multiple of 90, so the diagonal is swapped: the line now runs from top-left to bottom-right.

Note that 270 degrees clockwise is the same as 90 degrees anticlockwise, and either route gives the same answer. The line alone could not have determined the rotation, since a diagonal has rotational symmetry of order 2; the dot is what carries the information.

Question 7

A square sheet is folded in half vertically, then in half horizontally. A single hole is punched exactly on the vertical crease but away from the horizontal one. How many holes appear when unfolded, and where?

Accepted Answer

Two folds give 4 layers, so the naive count would be 4 holes. The punch position changes that.

Unfold the horizontal crease first, since it was folded last. The hole is reflected across the horizontal midline, giving two holes, one above and one below, both sitting on the vertical crease line.

Now unfold the vertical crease. Each existing hole should reflect across the vertical midline. But each hole lies exactly on that midline, so its reflection coincides with itself.

The two holes therefore do not double, and the final count is 2, not 4.

They sit on the vertical centre line of the sheet, symmetric about the horizontal centre line.

The general rule this illustrates is that a punch lying on a fold line halves the expected count for that fold, because the two layers it would have separated into are joined at the crease. A punch lying on both creases would give just 1 hole.

Question 8

A cube of side 5 units is painted on all six faces and cut into 125 unit cubes. How many unit cubes have at least one painted face, and how many have exactly one?

Accepted Answer

The fastest route to 'at least one' is the complement: total cubes minus the unpainted interior.

The interior is a cube of side n minus 2, which is 3, so it contains 3 cubed = 27 unpainted cubes.

At least one painted face: 125 minus 27 = 98.

For exactly one painted face, these are the face cubes excluding edges. Each of the 6 faces contributes an inner square of side n minus 2, so 6 times 3 squared = 6 times 9 = 54.

Verify with the full decomposition: corners 8, edges 12 times 3 = 36, faces 54, interior 27. The total is 8 + 36 + 54 + 27 = 125, which matches 5 cubed exactly.

The check also confirms the first answer, since 8 + 36 + 54 = 98. Performing the sum takes a few seconds and catches the commonest slip, which is using n rather than n minus 2 in the edge and face counts.

Question 9

A solid is built on a 3 by 3 base grid. The heights of the nine columns, read row by row from the back-left, are 3, 1, 2 / 1, 2, 1 / 2, 1, 3. How many unit cubes are used, and what does the front view look like?

Accepted Answer

Count the cubes by summing the column heights, since each column is a stack of that many unit cubes.

Row 1: 3 + 1 + 2 = 6. Row 2: 1 + 2 + 1 = 4. Row 3: 2 + 1 + 3 = 6.

Total: 16 unit cubes. Note that this count includes cubes hidden inside and behind the solid, which is why summing heights is safer than counting visible faces.

For the front view, look along the depth axis. Each of the three columns of the front view is as tall as the tallest column behind it in that position.

Column 1 contains heights 3, 1, 2, so the tallest is 3.
Column 2 contains heights 1, 2, 1, so the tallest is 2.
Column 3 contains heights 2, 1, 3, so the tallest is 3.

The front view is therefore a silhouette of heights 3, 2, 3.

The instructive point is that the front view discards a great deal: any solid whose column maxima are 3, 2, 3 produces exactly this view, and a 16-cube solid and an 8-cube solid can be indistinguishable from the front.

Practice — "Spatial Aptitude"

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