Chapter 11 Solutions — Exploring Some Geometric Themes

At Step 0, the shape is an equilateral triangle with 3 sides.

S₀ = 3

At each step, every existing side is replaced by 4 smaller segments (the middle third is replaced by two sides of a smaller equilateral triangle, giving a 'bump'). So the number of sides multiplies by 4 at each step.

S_{n+1} = 4 × Sₙ

Step 0: S₀ = 3

Step 1: S₁ = 4 × 3 = 12

Step 2: S₂ = 4 × 12 = 48

Step 3: S₃ = 4 × 48 = 192

General formula: Sₙ = 3 × 4^n

We examine the three given projection views and match them to one of the seven cube-stack solids shown (options i–vii).

Given views analysis:

• Front view: shows a C-shape (like a 2×2 block with the middle-right unit missing), indicating a step structure visible from the front.

• Top view: shows a staircase/L shape from above, meaning the solid has cubes arranged in a step pattern when seen from directly above.

• Side view: shows a small step block, consistent with a 2-step structure from the side.

Looking at all seven options:

• Option (iv) is a C-shaped arrangement of cubes (similar to the gold C-shape shown at the top right of the problem page). It produces:

— Front view: C / staircase shape ✓

— Top view: staircase/L arrangement ✓

— Side view: step/small block ✓

The other options produce different front, top, or side views that do not match all three simultaneously.

The image shows a triangular wall of cubes, one cube deep, viewed isometrically.

Reading it row by row from the top, each row down has one more cube than the row above:

Top row: 1 cube

2nd row: 2 cubes

3rd row: 3 cubes

Bottom row: 4 cubes

Total = 1 + 2 + 3 + 4 = 10 cubes.

This is the 4th triangular number, T₄ = 4 × 5 / 2 = 10 ✓