https://file.aiquickdraw.com/mathbot/file/1727011872304un4tc2kl.jpg

Let's analyze the patterns in each of the figures to find the value of \( y \) in Figure (III).

### Figure (I):
- Numbers: \( 2, 5, 1000, 1 \)
- If we examine the ratios:
  - \( 1000 \div 2 = 500 \)
  - \( 5 \div 1 = 5 \)
  - There isn't an obvious arithmetic progression, but if we use multiplication across diagonals:
  - \( 2 \times 1 = 2 \)
  - \( 5 \times 1000 = 5000 \)

### Figure (II):
- Numbers: \( 9, 10, 5000, 45 \)
- Diagonal products:
  - \( 9 \times 45 = 405 \)
  - \( 10 \times 5000 = 50000 \)

### Figure (III):
- Numbers: \( 5, 10, 1600, y \)
- Using the same diagonal multiplication logic:
  - \( 5 \times y = ? \)
  - \( 10 \times 1600 = 16000 \)

To solve for \( y \):
\[
5 \times y = 16000
\]
\[
y = \frac{16000}{5} = 3200
\]

Thus, the value of \( y \) is **3200**.

Let me know if you would like any further details or clarifications! Here are 5 questions to extend your understanding:
1. Can you find any other patterns in the figures that relate the numbers differently?
2. How would you approach solving this if there were no clear diagonal pattern?
3. Could this problem be solved using ratios or other mathematical operations?
4. How could changing one number affect the pattern in each figure?
5. What other kinds of mathematical puzzles involve this kind of diagonal multiplication?

**Tip:** Always try to look for patterns in multiple directions (rows, columns, diagonals) when faced with figure-based math puzzles!

What would replace 'y' in the figures given below?

Solve a math puzzle by finding 'y' in geometric figures using diagonal multiplication. This puzzle involves analyzing patterns in multiple figures, including multiplication across diagonals to find the unknown value y. Suitable for students in Grades 6-8.

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