Prime Time | A

Question 3

Make such pictures for the following:

a. 15 pegs, thread-gap of 10

b. 10 pegs, thread-gap of 7

c. 14 pegs, thread-gap of 6

d. 8 pegs, thread-gap of 3

Question diagram 1
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Solution
Understand the Question
  • To create thread patterns on a circle with NN numbered pegs:
    1. Start at peg 11.
    2. Add the thread-gap to find the next peg number.
    3. If the resulting number is greater than the total number of pegs NN, subtract NN from it.
    4. Continue this process until the thread loops back to peg 11.

a. 15 pegs, thread-gap of 10

Step 1 · Find Sequence of Connections

Given total pegs =15= 15 and thread-gap =10= 10.Diagram 1

Starting at peg 11: 1+10=111 + 10 = 11 11+10=21  ⟹  21−15=611 + 10 = 21 \implies 21 - 15 = 6 6+10=16  ⟹  16−15=16 + 10 = 16 \implies 16 - 15 = 1

Returning to starting peg 11, the sequence of connected pegs is: 1→11→6→11 \to 11 \to 6 \to 1

Answer

a. Connect pegs 1→11→6→11 \to 11 \to 6 \to 1 to form a triangle.

b. 10 pegs, thread-gap of 7

Step 1 · Find Sequence of Connections

Given total pegs =10= 10 and thread-gap =7= 7.Diagram 2

Starting at peg 11, the sequence of connections is:

1→(1+7)=88→(8+7)=15→(15−10)=55→(5+7)=12→(12−10)=22→(2+7)=99→(9+7)=16→(16−10)=66→(6+7)=13→(13−10)=33→(3+7)=1010→(10+7)=17→(17−10)=77→(7+7)=14→(14−10)=44→(4+7)=11→(11−10)=1\begin{aligned} 1 &\to (1 + 7) = 8 \\ 8 &\to (8 + 7) = 15 \to (15 - 10) = 5 \\ 5 &\to (5 + 7) = 12 \to (12 - 10) = 2 \\ 2 &\to (2 + 7) = 9 \\ 9 &\to (9 + 7) = 16 \to (16 - 10) = 6 \\ 6 &\to (6 + 7) = 13 \to (13 - 10) = 3 \\ 3 &\to (3 + 7) = 10 \\ 10 &\to (10 + 7) = 17 \to (17 - 10) = 7 \\ 7 &\to (7 + 7) = 14 \to (14 - 10) = 4 \\ 4 &\to (4 + 7) = 11 \to (11 - 10) = 1 \end{aligned}

Returning to starting peg 11, the sequence of connected pegs is: 1→8→5→2→9→6→3→10→7→4→11 \to 8 \to 5 \to 2 \to 9 \to 6 \to 3 \to 10 \to 7 \to 4 \to 1

Answer

b. Connect pegs 1→8→5→2→9→6→3→10→7→4→11 \to 8 \to 5 \to 2 \to 9 \to 6 \to 3 \to 10 \to 7 \to 4 \to 1 to form a 10-pointed star.

c. 14 pegs, thread-gap of 6

Step 1 · Find Sequence of Connections

Given total pegs =14= 14 and thread-gap =6= 6.Diagram 3

Starting at peg 11, the sequence of connections is:

1→(1+6)=77→(7+6)=1313→(13+6)=19→(19−14)=55→(5+6)=1111→(11+6)=17→(17−14)=33→(3+6)=99→(9+6)=15→(15−14)=1\begin{aligned} 1 &\to (1 + 6) = 7 \\ 7 &\to (7 + 6) = 13 \\ 13 &\to (13 + 6) = 19 \to (19 - 14) = 5 \\ 5 &\to (5 + 6) = 11 \\ 11 &\to (11 + 6) = 17 \to (17 - 14) = 3 \\ 3 &\to (3 + 6) = 9 \\ 9 &\to (9 + 6) = 15 \to (15 - 14) = 1 \end{aligned}

Returning to starting peg 11, the sequence of connected pegs is: 1→7→13→5→11→3→9→11 \to 7 \to 13 \to 5 \to 11 \to 3 \to 9 \to 1

Answer

c. Connect pegs 1→7→13→5→11→3→9→11 \to 7 \to 13 \to 5 \to 11 \to 3 \to 9 \to 1 to form a 7-pointed star.

d. 8 pegs, thread-gap of 3

Step 1 · Find Sequence of Connections

Given total pegs =8= 8 and thread-gap =3= 3.Diagram 4

Starting at peg 11, the sequence of connections is:

1→(1+3)=44→(4+3)=77→(7+3)=10→(10−8)=22→(2+3)=55→(5+3)=88→(8+3)=11→(11−8)=33→(3+3)=66→(6+3)=9→(9−8)=1\begin{aligned} 1 &\to (1 + 3) = 4 \\ 4 &\to (4 + 3) = 7 \\ 7 &\to (7 + 3) = 10 \to (10 - 8) = 2 \\ 2 &\to (2 + 3) = 5 \\ 5 &\to (5 + 3) = 8 \\ 8 &\to (8 + 3) = 11 \to (11 - 8) = 3 \\ 3 &\to (3 + 3) = 6 \\ 6 &\to (6 + 3) = 9 \to (9 - 8) = 1 \end{aligned}

Returning to starting peg 11, the sequence of connected pegs is: 1→4→7→2→5→8→3→6→11 \to 4 \to 7 \to 2 \to 5 \to 8 \to 3 \to 6 \to 1

Answer

d. Connect pegs 1→4→7→2→5→8→3→6→11 \to 4 \to 7 \to 2 \to 5 \to 8 \to 3 \to 6 \to 1 to form an 8-pointed star.

Common Mistakes
  • Subtraction Error: Forgetting to subtract the total number of pegs whenever the sum exceeds the total peg count.
  • Premature Termination: Stopping the connection sequence before returning all the way to the initial starting peg (11).
  • Step Count Misinterpretation: Adding thread-gap from 00 instead of adding it to the current peg's number.

More questions in A

Q1

Let us now play the 'idli-vada' game with different pairs of numbers:

(a) 2 and 5, (b) 3 and 7, (c) 4 and 6.

We will say 'idli' for multiples of the smaller number, 'vada' for multiples of the larger number and 'idli-vada' for common multiples. Draw a figure similar to Fig. 5.1 if the game is played up to 60.

Q2

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?

Q3

Make such pictures for the following:

a. 15 pegs, thread-gap of 10

b. 10 pegs, thread-gap of 7

c. 14 pegs, thread-gap of 6

d. 8 pegs, thread-gap of 3

Q4

Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. Share with your classmates and find out who else gave the same reasons as you did. Did anyone give different reasons that may not have occurred to you?

Q5

A prime puzzle

The figure on the left shows the puzzle. The figure on the right shows the solution of the puzzle. Think what the rules can be to solve the puzzle.

Q6

Rules

Fill the grid with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

Q7

Q. Solve the prime puzzles by filling the grids with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

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