Let’s solve the puzzles step-by-step:

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Puzzle 4:

The triangular number puzzle has:

Row 1: ?
Row 2: 9, 25
Row 3: 4, 36
Row 4: 1, 81, 64, 49

Observing the pattern:

• Each number in the rows represents squared numbers.
• Row 4: $1^2, 9^2, 8^2, 7^2$ (1, 81, 64, 49)
• Row 3: $2^2, 6^2$ (4, 36)
• Row 2: $3^2, 5^2$ (9, 25)
• Row 1 should logically be $4^2 = 16$.

Answer: (a) 16

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Puzzle 5:

The numbers inside triangles are:

3 → 5 → 8 → 13 → 22 → ?

This is a sequence where the difference between consecutive numbers increases progressively:

• $5 - 3 = 2$
• $8 - 5 = 3$
• $13 - 8 = 5$
• $22 - 13 = 9$

Next difference will be $9 + 2 = 11$:

• $22 + 11 = 33$

Answer: (a) 33

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Puzzle 6:

The grid is:

2   8   9
3   2   4
3   6   ?

The pattern relates to the multiplication of the first two numbers in each row:

• Row 1: $2 \times 4 = 8$, $8 + 1 = 9$
• Row 2: $3 \times 2 = 6$, $6 + (-2) = 4$
• Row 3: $3 \times 2 = 6$, $6 + (-5) = 1$

So, the missing number is 1.

Answer: (a) 1

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Final Answers:

1. Puzzle 4: (a) 16
2. Puzzle 5: (a) 33
3. Puzzle 6: (a) 1

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Do you have any specific doubts about these? Would you like a detailed breakdown of any step?

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5 Related Questions:

1. How do number patterns often relate to squares or cubes?
2. What are other common sequence patterns found in puzzles like these?
3. How does one identify arithmetic vs geometric progressions in puzzles?
4. Can there be alternative ways to solve Puzzle 6?
5. Why is identifying differences in sequences critical in solving puzzles?

Tip: Always check for square, cube, or arithmetic progression patterns when working on number puzzles. They are common tricks in such problems!