https://file.aiquickdraw.com/mathbot/file/17216290887128mrs310h.jpg

Let's reanalyze questions 8 and 9 using the clearer image provided.

### Question 8
The uniform step method \( \begin{bmatrix} a & 5 & 2 \\ b & 1 & 3 \end{bmatrix} \) is used to fill a \( 7 \times 7 \) square with numbers \( 0, \ldots, 48 \). If \( 0 \) is placed in the upper right-hand corner of the square, what is the element in the lower left-hand corner?

To determine the value in the lower left-hand corner, we need to understand how the uniform step method works. The uniform step method involves a specific pattern for filling the square. Here, the upper right-hand corner has 0, and we need to trace the filling process through the given matrix to find the value in the lower left-hand corner.

However, without specific steps or a detailed pattern of filling the matrix, it's not possible to determine the exact value for the lower left-hand corner. We need more information about how \( a \) and \( b \) influence the filling pattern.

### Question 9
As in the previous question, the uniform step method \( \begin{bmatrix} a & 5 & 2 \\ b & 1 & 3 \end{bmatrix} \) is used to fill a \( 7 \times 7 \) square with numbers \( 0, \ldots, 48 \) and 0 is placed in the upper right-hand corner.

Options:
1. **The square is magic.**
2. **The square is filled.**
3. **The square is diabolic and the diabolic sum is 168.**
4. **One of the rows of the square is \([36, 47, 2, 13, 17, 20, 33]\).**

From the previous analysis:
1. **The square is magic:** A magic square is one where the sums of numbers in each row, column, and diagonal are equal. The uniform step method aims to create such squares, so this is likely true if correctly applied.
2. **The square is filled:** Since it fills a \( 7 \times 7 \) square with numbers \( 0 \) to \( 48 \), this should be true.
3. **The square is diabolic and the diabolic sum is 168:** This depends on the specific properties of the sum in this particular uniform step method application. Typically, if the square is not specifically designed to be diabolic, this could be false.
4. **One of the rows of the square is \([36, 47, 2, 13, 17, 20, 33]\):** Given the previous row sum calculation, this is likely true.

### Final Answers for Questions 8 and 9:
1. **Question 8:** Without additional steps or pattern rules, we cannot determine the exact value for the lower left-hand corner.
2. **Question 9:** Based on typical properties of uniform step methods and magic squares:
   - **The square is magic:** True
   - **The square is filled:** True
   - **The square is diabolic and the diabolic sum is 168:** False
   - **One of the rows of the square is \([36, 47, 2, 13, 17, 20, 33]\):** True

Would you like further clarifications or have any more questions? Here are five more questions you might find interesting:
1. What specific rules govern the uniform step method for filling squares?
2. How can we verify if a square is diabolic?
3. What are the differences between standard magic squares and diabolic magic squares?
4. How do we determine the specific pattern used in the uniform step method?
5. Can you provide examples of filled \( 7 \times 7 \) magic squares using different methods?

**Tip:** When dealing with structured methods like the uniform step method, understanding the step-by-step process and pattern is crucial for determining specific elements in the matrix.

Explore the uniform step method used in filling magic squares, focusing on specific examples and analysis. Learn how this method applies to creating magic squares and understand its properties.

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