Patterns in Mathematics | FIO

Question 22

Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

Solution

Understand the Question

A complete graph with $n$ points (vertices) connects every point to every other point with a straight line (edge).
Counting the lines in complete graphs with $2, 3, 4, 5,$ and $6$ points gives a sequence of numbers called triangular numbers.
Since each of the $n$ points connects to the remaining $(n - 1)$ points and every line connects $2$ points (meaning each line is counted twice), the total number of lines is given by the formula: $\text{Number of lines} = \dfrac{n \times (n - 1)}{2}$

Step 1 · Count lines in each complete graph

Diagram 1

Counting the lines in each complete graph:

For $K_2$ ( $2$ points): $1$ line
For $K_3$ ( $3$ points): $3$ lines
For $K_4$ ( $4$ points): $6$ lines
For $K_5$ ( $5$ points): $10$ lines
For $K_6$ ( $6$ points): $15$ lines

Step 2 · Identify the number sequence

The number of lines counted gives the sequence: $1, 3, 6, 10, 15, \dots$

This is the sequence of triangular numbers, so named because these quantities of dots can be arranged in the shape of equilateral triangles:

$1$ (a single dot)
$3 = 1 + 2$ (base of $2$ dots)
$6 = 1 + 2 + 3$ (base of $3$ dots)
$10 = 1 + 2 + 3 + 4$ (base of $4$ dots)
$15 = 1 + 2 + 3 + 4 + 5$ (base of $5$ dots)

Step 3 · Explain the formula and verify for each graph

In a complete graph with $n$ points, each point connects to every other point ( $n - 1$ points).

Multiplying gives $n \times (n - 1)$ connections. Since each line connects two endpoints, every line is counted twice (e.g., from point A to B and from B to A). Therefore, we divide by $2$ :

$\text{Number of lines} = \dfrac{n \times (n - 1)}{2}$

Verifying for each complete graph:

For $K_2$ ( $n = 2$ ):

\begin{aligned} \text{Number of lines} &= \dfrac{2 \times (2 - 1)}{2} \\[0.6em] &= \dfrac{2 \times 1}{2} \\[0.6em] &= \dfrac{2}{2} = 1 \end{aligned}

For $K_3$ ( $n = 3$ ):

\begin{aligned} \text{Number of lines} &= \dfrac{3 \times (3 - 1)}{2} \\[0.6em] &= \dfrac{3 \times 2}{2} \\[0.6em] &= \dfrac{6}{2} = 3 \end{aligned}

For $K_4$ ( $n = 4$ ):

\begin{aligned} \text{Number of lines} &= \dfrac{4 \times (4 - 1)}{2} \\[0.6em] &= \dfrac{4 \times 3}{2} \\[0.6em] &= \dfrac{12}{2} = 6 \end{aligned}

For $K_5$ ( $n = 5$ ):

\begin{aligned} \text{Number of lines} &= \dfrac{5 \times (5 - 1)}{2} \\[0.6em] &= \dfrac{5 \times 4}{2} \\[0.6em] &= \dfrac{20}{2} = 10 \end{aligned}

For $K_6$ ( $n = 6$ ):

\begin{aligned} \text{Number of lines} &= \dfrac{6 \times (6 - 1)}{2} \\[0.6em] &= \dfrac{6 \times 5}{2} \\[0.6em] &= \dfrac{30}{2} = 15 \end{aligned}

Answer

Number of lines: $1, 3, 6, 10, 15$

Sequence name: Triangular numbers
Reason: In a complete graph with $n$ points, each point connects to $(n - 1)$ other points. Dividing by $2$ to avoid counting each line twice gives the formula $\dfrac{n(n - 1)}{2}$ .

Common Mistakes

Double-Counting Error: Forgetting to divide by $2$ , which gives $n(n - 1)$ lines instead of $\dfrac{n(n - 1)}{2}$ .
Confusing Points and Lines: Stating $n$ instead of the number of line segments between the points.

More questions in FIO

Can you think of other examples where mathematics helps us in our everyday lives?

How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)

Can you recognise the pattern in each of the sequences in Table 1?

Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.

Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!

Why are $1, 3, 6, 10, 15, \dots$ called triangular numbers? Why are $1, 4, 9, 16, 25, \dots$ called square numbers or squares? Why are $1, 8, 27, 64, 125, \dots$ called cubes?

You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this!

This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!

What would you call the following sequence of numbers?

That's right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?

Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?

Q10

Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., $1$ , $1 + 2 + 1$ , $1 + 2 + 3 + 2 + 1$ , $\dots$ , gives square numbers?

Q11

By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of $1 + 2 + 3 + \dots + 99 + 100 + 99 + \dots + 3 + 2 + 1$ ?

Q12

Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down?

Q13

Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?

Q14

What happens when you add up pairs of consecutive triangular numbers? That is, take $1 + 3, 3 + 6, 6 + 10, 10 + 15, \dots$ Which sequence do you get? Why? Can you explain it with a picture?

Q15

What happens when you start to add up powers of $2$ starting with $1$ , i.e., take $1$ , $1 + 2$ , $1 + 2 + 4$ , $1 + 2 + 4 + 8, \dots$ ? Now add $1$ to each of these numbers—what numbers do you get? Why does this happen?

Q16

What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?

Q17

What happens when you start to add up hexagonal numbers, i.e., take $1$ , $1 + 7$ , $1 + 7 + 19$ , $1 + 7 + 19 + 37, \dots$ ? Which sequence do you get? Can you explain it using a picture of a cube?

Question diagram(s):

Q18

Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?

Q19

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

Q20

Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.

Q21

Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?

Q22

Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

Q23

How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?

Q24

How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)

Q25

To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '—' by a 'speed bump' _/_ . As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is $3, 12, 48, \dots$ , i.e., $3$ times Powers of $4$ ; this sequence is not shown in Table 1.)