Can you recognise the pattern in each of the sequences in Table 3?

Question

Homi · Accepted Answer

Understand the Question

Each row in Table 3 shows a sequence of geometric shapes that follows a specific growth rule from one step to the next.
To identify the pattern in each sequence, examine how the number of sides, vertices, component shapes, or subdivisions increases with each successive figure.

Step 1 · Regular Polygons

Diagram 1

Each shape is a regular polygon (all sides and interior angles are equal).
The number of sides increases by $1$ $1$ at each step:
- Triangle: $3$ sides
- Quadrilateral: $4$ sides
- Pentagon: $5$ sides
- Continuing up to a decagon with $10$ sides.

Step 2 · Complete Graphs

Diagram 2

Each graph consists of vertices (points) where every vertex is connected to every other vertex by an edge.
The number of vertices increases by $1$ $1$ at each step:
- $K_2$ : $2$ vertices
- $K_3$ : $3$ vertices
- $K_4$ : $4$ vertices
- Continuing up to $K_6$ with $6$ vertices.

Step 3 · Stacked Squares

Diagram 3

Larger squares are constructed using smaller unit squares.
The side length increases by $1$ $1$ unit square at each step:
- $1 \times 1$ square ( $1$ small square)
- $2 \times 2$ square ( $4$ small squares)
- $3 \times 3$ square ( $9$ small squares)
The total number of small squares in step $n$ is $n^2$ .

Step 4 · Stacked Triangles

Diagram 4

Larger equilateral triangles are built from smaller unit triangles.
The base length increases by $1$ $1$ unit triangle at each step:
- Base of $1$ triangle $\implies 1$ unit triangle
- Base of $2$ triangles $\implies 4$ unit triangles
- Base of $3$ triangles $\implies 9$ unit triangles
The total number of unit triangles in step $n$ is $n^2$ .

Step 5 · Koch Snowflake

Diagram 5

Starts with an equilateral triangle.
At each subsequent stage, the middle third of every straight line segment is replaced by an outward-pointing smaller equilateral triangle.
Repeating this process iteratively produces the fractal Koch snowflake.

Answer

Yes, we can recognise the pattern in each sequence of Table 3:

Regular Polygons: The number of sides increases by $1$ at each step (from $3$ up to $10$ ).
Complete Graphs ( $K_n$ ): The number of vertices increases by $1$ at each step, with every vertex connected to all other vertices.
Stacked Squares: The side length increases by $1$ unit square, giving total squares $= n^2$ .
Stacked Triangles: The base length increases by $1$ unit triangle, giving total unit triangles $= n^2$ .
Koch Snowflake: An equilateral triangle is added to the middle third of each line segment iteratively.

Common Mistakes

Counting total shapes vs. side length: For stacked squares and stacked triangles, the total number of unit shapes grows quadratically ( $n^2$ ), not by $+1$ .
Confusing Complete Graphs with Polygons: In complete graphs ( $K_n$ ), edges connect all pairs of vertices (including diagonals), whereas polygons only connect adjacent vertices along the perimeter.

Question 19