Solution: Figure It Out 1.6

Section 1.6: Relation to Number Sequences

Q1. Count the number of sides and corners in Regular Polygons.

Answer:

• Sides: $3, 4, 5, 6, 7, 8, 9, 10 \dots$
• Corners: $3, 4, 5, 6, 7, 8, 9, 10 \dots$
• Conclusion: In any closed polygon, Number of Sides = Number of Corners.

Q2. Count the number of lines in Complete Graphs.

Answer:

• The sequence is 1, 3, 6, 10, 15…
• This is the Triangular Number sequence.
• Why? To draw a complete graph with $n$ points, each new point connects to all previous points. When you add the 3rd point, you draw 2 new lines. When you add the 4th point, you draw 3 new lines. This is adding counting numbers ($1 + 2 + 3 + \dots$).

Q3. How many little squares are in the Stacked Squares sequence?

Answer:

• Sequence: 1, 4, 9, 16, 25…
• This is the Square Number sequence.

Q4. How many little triangles are in the Stacked Triangles sequence?

Answer:

• Sequence: 1, 4, 9, 16, 25…
• This is also the Square Number sequence!
• Explanation: In row 1, there is 1 triangle. In row 2, there are 3 triangles. In row 3, there are 5. Summing these rows ($1 + 3 + 5$) is the “Sum of Odd Numbers” rule, which creates Square Numbers.

Q5. How many total line segments are there in each shape of the Koch Snowflake?

Answer:

• Shape 1 (Triangle): 3 segments.
• Shape 2: Each of the 3 segments gets a “bump”, turning 1 segment into 4. Total = $3 \times 4 = 12$.
• Shape 3: Each of the 12 segments turns into 4. Total = $12 \times 4 = 48$.
• Sequence: $3, 12, 48, 192, 768 \dots$