To determine the missing number, let's analyze the given sequences and search for a repeating algebraic equation involving basic arithmetic operations.

Given sequences:

1. $2, -3, 9, 5$
2. $8, ?, 3, 7$
3. $-4, 2, 4, -6$
4. $5, 2, 8, 1$

Step-by-step analysis:

Let's examine each column across the rows to see if there’s a common pattern:

First Column: $2, 8, -4, 5$

Look at the differences:

• $8 - 2 = 6$
• $-4 - 8 = -12$
• $5 - (-4) = 9$

There seems to be no consistent arithmetic relationship in the first column, so let's try looking at the other columns.

Second Column: $-3, ?, 2, 2$

This column is missing a number, but let's observe other columns first before returning to this.

Third Column: $9, 3, 4, 8$

Look at the differences:

• $3 - 9 = -6$
• $4 - 3 = 1$
• $8 - 4 = 4$

While there is no immediate linear relationship here, let's check the fourth column.

Fourth Column: $5, 7, -6, 1$

Look at the differences:

• $7 - 5 = 2$
• $-6 - 7 = -13$
• $1 - (-6) = 7$

Still, no consistent arithmetic pattern is emerging across any of the columns directly. Thus, it’s likely that the algebraic rule involves some operation that combines elements from multiple columns.

Hypothesis:

There could be a relationship involving sums or operations between adjacent columns.

To summarize: The analysis above suggests a complex pattern that may not directly follow simple column-wise arithmetic progressions.

Since I need to refine this, let me test