Question 3

We will draw patterns by connecting pegs on a circle.

Step 1 — Understanding the pattern rule

We have a circle with numbered pegs. We start at a peg, usually peg 1. The "thread-gap" tells us how many pegs to jump. Connect the current peg to the next peg. The next peg is "thread-gap" steps away, clockwise. If the next peg number is too high, subtract the total. Keep connecting until we reach our starting peg.

Step 2 — Drawing for 15 pegs, thread-gap of 10

We have 15 pegs on the circle. The thread-gap is 10. Let us start at peg 1. The next peg is: $1 + 10 = 11$ The next peg is: $11 + 10 = 21$ Since 21 is more than 15, we subtract 15: $21 - 15 = 6$ The next peg is: $6 + 10 = 16$ Since 16 is more than 15, we subtract 15: $16 - 15 = 1$ We are back to peg 1. So, we connect pegs 1 to 11, then 11 to 6, then 6 to 1.

Step 3 — Drawing for 10 pegs, thread-gap of 7

We have 10 pegs on the circle. The thread-gap is 7. Let us start at peg 1. The sequence of connections is: $1 \rightarrow (1+7) = 8$ $8 \rightarrow (8+7) = 15 \rightarrow (15-10) = 5$ $5 \rightarrow (5+7) = 12 \rightarrow (12-10) = 2$ $2 \rightarrow (2+7) = 9$ $9 \rightarrow (9+7) = 16 \rightarrow (16-10) = 6$ $6 \rightarrow (6+7) = 13 \rightarrow (13-10) = 3$ $3 \rightarrow (3+7) = 10$ $10 \rightarrow (10+7) = 17 \rightarrow (17-10) = 7$ $7 \rightarrow (7+7) = 14 \rightarrow (14-10) = 4$ $4 \rightarrow (4+7) = 11 \rightarrow (11-10) = 1$ We are back to peg 1. We connect pegs in this order: 1-8-5-2-9-6-3-10-7-4-1.

Step 4 — Drawing for 14 pegs, thread-gap of 6

We have 14 pegs on the circle. The thread-gap is 6. Let us start at peg 1. The sequence of connections is: $1 \rightarrow (1+6) = 7$ $7 \rightarrow (7+6) = 13$ $13 \rightarrow (13+6) = 19 \rightarrow (19-14) = 5$ $5 \rightarrow (5+6) = 11$ $11 \rightarrow (11+6) = 17 \rightarrow (17-14) = 3$ $3 \rightarrow (3+6) = 9$ $9 \rightarrow (9+6) = 15 \rightarrow (15-14) = 1$ We are back to peg 1. We connect pegs in this order: 1-7-13-5-11-3-9-1.

Step 5 — Drawing for 8 pegs, thread-gap of 3

We have 8 pegs on the circle. The thread-gap is 3. Let us start at peg 1. The sequence of connections is: $1 \rightarrow (1+3) = 4$ $4 \rightarrow (4+3) = 7$ $7 \rightarrow (7+3) = 10 \rightarrow (10-8) = 2$ $2 \rightarrow (2+3) = 5$ $5 \rightarrow (5+3) = 8$ $8 \rightarrow (8+3) = 11 \rightarrow (11-8) = 3$ $3 \rightarrow (3+3) = 6$ $6 \rightarrow (6+3) = 9 \rightarrow (9-8) = 1$ We are back to peg 1. We connect pegs in this order: 1-4-7-2-5-8-3-6-1.

Answer

(a) Connect pegs 1-11, 11-6, 6-1 to form a triangle. (b) Connect pegs 1-8-5-2-9-6-3-10-7-4-1 to form a 10-pointed star. (c) Connect pegs 1-7-13-5-11-3-9-1 to form a 7-pointed star. (d) Connect pegs 1-4-7-2-5-8-3-6-1 to form an 8-pointed star.