14.4 3D Views, Rotation & Hidden Cubes

3. Orthogonal Views & 3D Block Visualisation

In addition to cube folding, 3D Spatial Reasoning on the Pre-Test includes questions evaluating composite block models constructed from unit cubes. Candidates are tested on:

1. Orthogonal Projections: Converting a 3D block model into its 2D Plan View (top-down), Front Elevation (front-facing), and Side Elevation (side-facing).
2. Hidden (Concealed) Block Counting: Determining the exact total number of unit cubes required to build a solid structure, including cubes hidden from view.

                      ORTHOGONAL VIEW PERSPECTIVES

                           PLAN VIEW (Top-Down)
                                   │
                                   ▼
                             ┌───┬───┬───┐
                             │   │   │   │
       SIDE ELEVATION ──────►│ 3D BLOCK  │◄────── SIDE ELEVATION
       (Left View)           │ STRUCTURE │        (Right View)
                             └───┴───┴───┘
                                   ▲
                                   │
                         FRONT ELEVATION (Front View)

Method for Counting Hidden Blocks: The Column Height Grid

When viewing an isometric diagram of stacked blocks:

1. Read the support condition: When the problem states there are solid stacks with no gaps, a top cube at height 3 has cubes at levels 1 and 2 beneath it. Do not assume a solid column if bridges, hollow spaces or other support arrangements are allowed.
2. The Column Height Grid:
   • Draw or visualize a 2D top-down grid matching the footprint of the structure.
   • In each grid cell, write the exact height (number of cubes stacked) in that vertical column.
   • Sum all column numbers to calculate the total cube count.

With solid columns and no gaps, this specified model contains 3 + 1 + 2 + 2 = 8 cubes. A plain plan showing only occupied cells would not supply those four heights.

---

Counting a specified solid

Worked Example 3: Hidden Block Counting in a 3D Block Structure

Problem Statement: A child builds a stepped staircase structure from identical unit cubes. The front row has a column of 1 cube on the left, a column of 2 cubes in the middle, and a column of 3 cubes on the right. Directly behind this front row sits a second row of columns: behind the 1-cube column sits a 2-cube column; behind the 2-cube column sits a 3-cube column; behind the 3-cube column sits a 4-cube column. Assuming the structure is solid with no hollow air pockets underneath, how many total unit cubes are in the structure, and how many are completely hidden from a direct front orthographic view (looking straight along the rows, with no top or side faces counted)?

Step-by-Step Solution Breakdown:

1. Map the Column Heights on a 2x3 Grid:
   • Front Row: $[1, 2, 3]$
   • Back Row: $[2, 3, 4]$
2. Calculate Total Unit Cubes: $$\text{Total Cubes} = (1 + 2 + 3) + (2 + 3 + 4) = 6 + 9 = \mathbf{15 \text{ cubes}}$$
3. Identify Completely Hidden Cubes:
   • In the back row, the column of height 2 sits directly behind the front column of height 1. The bottom cube of this back column is obscured by the front cube; the top cube (cube 2) is visible above the front column in the front elevation.
   • In the back row, the column of height 3 sits behind the front column of height 2. Its bottom 2 cubes are obscured; its top cube (cube 3) is visible.
   • In the back row, the column of height 4 sits behind the front column of height 3. Its bottom 3 cubes are obscured; its top cube (cube 4) is visible.
   • Therefore, the concealed cubes hidden entirely behind taller or equal front cubes are $1 + 2 + 3 = \mathbf{6 \text{ concealed cubes}}$.

Rotate a solid and read its views

A plan view looks straight down. A front elevation looks horizontally from the named front, while a side elevation looks horizontally from the named side. These are different from a corner picture showing several faces at once. Mark the front on your sketch before comparing views; changing the front can exchange the order of columns.

For a model of solid vertical stacks, write the height of each column in its floor cell. Consider this footprint:

The plan footprint contains six occupied cells. The front elevation has heights 2, 3, 4, taking the greatest height in each left-to-right column because the taller back stack can be seen above the front stack. The side elevation has two depth positions: the front-row maximum is 3 and the back-row maximum is 4. State the viewing side before drawing these positions from left to right. The total count is the sum of all heights: 15 cubes, not the sum of the elevation heights.

Now mentally turn the complete solid a quarter-turn on the table. The floor cells and their heights move together; cubes are not added or removed. A plan with six cells remains a six-cell footprint after the turn, and the total remains 15. What changes is which stacks line up behind one another from the fixed observer's position. This explains why a front elevation can change even though the object itself has the same number of cubes.

Views may not determine a unique model

Suppose a 2×2 footprint has every cell occupied. Models with back/front height rows [2,1]/[1,2] and [2,2]/[2,2] both have front maximum heights [2,2] and side maximum heights [2,2], yet contain 6 and 8 cubes respectively. Their plain occupied-cell plans also match. Thus plan, front and side outlines alone need not reveal every hidden height. A height-labelled plan gives more information than an ordinary plan outline.

If an exercise asks for a minimum, build the fewest cubes consistent with every stated view and support condition. If it asks for an exact count, check that the drawing or wording actually fixes the concealed columns. Do not fill every hidden position merely because one possible solid model would do so.

Keep 3D rotations separate from reflections

Turning a cube moves its faces as a rigid object. Adjacent faces remain adjacent, opposite faces remain opposite and an arrow stays attached to its own face. In the cross net earlier, hold face C as the front, fold A up, F down, B left and D right; E closes the back. A rightward arrow on C points toward D. After turning the entire cube, its screen direction may change, but it still points toward the edge it shares with D.

For labelled-face puzzles, first check the three opposite pairs, then the order around the visible corner, then the directions of any arrows. A drawing can pass the opposite-pair check and still have two side faces exchanged in a way that mirrors the solid. Use the actual net to trace the order instead of treating three mutually adjacent face names as proof of one particular pictured orientation.