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Question 2

Co-prime art

Observe the following thread art. The first diagram has 12 pegs and the thread is tied to every fourth peg (we say that the thread-gap is 4). The second diagram has 13 pegs and the thread-gap is 3. What about the other diagrams? Observe these pictures, share and discuss your findings in class.

In some diagrams, the thread is tied to every peg. In some, it is not. Is it related to the two numbers (the number of pegs and the thread-gap) being co-prime?

Solution

Understand the Question

Thread-gap is the number of steps the thread advances from one peg to the next around the circle.
Two numbers are co-prime if their Highest Common Factor ( $\text{HCF}$ ) is $1$ .
The thread visits every single peg on the circle if and only if the total number of pegs and the thread-gap are co-prime ( $\text{HCF} = 1$ ). If $\text{HCF} > 1$ , the thread cycles back to the starting peg without visiting all pegs.

Step 1 · Analyze the First Diagram (12 Pegs, Gap 4)

Diagram 1

Number of pegs: $12$
Thread-gap: $4$ (starts at $12$ , jumps $4$ steps to $4$ , then $8$ , then back to $12$ )
Pegs touched: Only pegs $12, 4,$ and $8$ (does not touch every peg)

Finding the common factors: $\text{Factors of } 12 = 1, 2, 3, 4, 6, 12$ $\text{Factors of } 4 = 1, 2, 4$ $\text{Highest Common Factor } (\text{HCF}) = 4$

Step 2 · Analyze the Second Diagram (13 Pegs, Gap 3)

Diagram 2

Number of pegs: $13$
Thread-gap: $3$ (starts at $13$ , jumps $3$ steps to $3$ , then $6, 9, \dots$ )
Pegs touched: Touches every peg

Finding the common factors: $\text{Factors of } 13 = 1, 13$ $\text{Factors of } 3 = 1, 3$ $\text{Highest Common Factor } (\text{HCF}) = 1$

Step 3 · Analyze the Third Diagram (16 Pegs, Gap 6)

Diagram 3

Number of pegs: $16$
Thread-gap: $6$ (starts at $16$ , jumps $6$ steps to $6$ , then $12, 2, \dots$ )
Pegs touched: Misses odd pegs like $1, 3, 5$ (does not touch every peg)

Finding the common factors: $\text{Factors of } 16 = 1, 2, 4, 8, 16$ $\text{Factors of } 6 = 1, 2, 3, 6$ $\text{Highest Common Factor } (\text{HCF}) = 2$

Step 4 · Analyze the Fourth Diagram (24 Pegs, Gap 6)

Diagram 4

Number of pegs: $24$
Thread-gap: $6$ (starts at $24$ , jumps $6$ steps to $6$ , then $12, 18,$ and $24$ )
Pegs touched: Only pegs $24, 6, 12,$ and $18$ (does not touch every peg)

Finding the common factors: $\text{Factors of } 24 = 1, 2, 3, 4, 6, 8, 12, 24$ $\text{Factors of } 6 = 1, 2, 3, 6$ $\text{Highest Common Factor } (\text{HCF}) = 6$

Step 5 · Determine the Co-prime Relationship

Comparing the four diagrams:

\begin{array}{|c|c|c|c|c|} \hline \text{Diagram} & \text{Pegs } (n) & \text{Gap } (g) & \text{HCF}(n, g) & \text{Touches Every Peg?} \\ \hline 1 & 12 & 4 & 4 & \text{No} \\ \hline 2 & 13 & 3 & 1 & \text{Yes (Co-prime)} \\ \hline 3 & 16 & 6 & 2 & \text{No} \\ \hline 4 & 24 & 6 & 6 & \text{No} \\ \hline \end{array}

When $\text{HCF}(\text{pegs}, \text{gap}) = 1$ (i.e. they are co-prime), the thread visits every peg.
When $\text{HCF}(\text{pegs}, \text{gap}) > 1$ (not co-prime), the thread forms a smaller closed loop and does not touch every peg.

Answer

The third diagram has 16 pegs with a thread-gap of 6.

The fourth diagram has 24 pegs with a thread-gap of 6.
Yes, the thread is tied to every peg if and only if the number of pegs and the thread-gap are co-prime.

Common Mistakes

Confusing Prime with Co-prime: The number of pegs does not need to be a prime number itself (e.g. $15$ pegs with gap $4$ will visit all pegs because $\text{HCF}(15, 4) = 1$ ). They only need to be co-prime to each other.
Miscounting the Thread-gap: Counting the gap starting from $1$ instead of counting the number of step intervals from the starting peg.

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