Explanation:
Okay, so I need to figure out how many integer coordinate points are strictly inside this rectangle with vertices at (5,4), (-5,4), (-5,-4), and (5,-4). Let me start by visualizing the rectangle. The rectangle spans from x = -5 to x = 5 and y = -4 to y = 4. But since the question is about points strictly inside, I need to exclude the boundaries.

First, let me recall that for a rectangle with sides parallel to the axes, the number of interior points can be calculated by considering the number of integer points along the x-axis and y-axis, then multiplying them. But since we need strictly interior points, we have to adjust for the boundaries.

Let me think. If the rectangle goes from x = -5 to x = 5, then the length along the x-axis is from -5 to 5. But since we want strictly interior points, we need to exclude the edges. So, for the x-coordinates, the interior points would be from -4 to 4, right? Because if you have a rectangle from -5 to 5, the interior would be points where x is greater than -5 and less than 5. But since we are dealing with integer coordinates, the integer points strictly inside would have x-values from -4 up to 4. Similarly for the y-coordinates, the original rectangle goes from -4 to 4, so the interior would be y-values from -3 up to 3.

Wait, let me check that. If the rectangle includes the boundary points, then the total number of integer points on the boundary would be... but actually, the question is about strictly interior points. So, for the x-coordinate, the original rectangle includes x = -5 and x = 5. So, to be strictly inside, x must be greater than -5 and less than 5. Since x is integer, that means x can be from -4, -3, -2, -1, 0, 1, 2, 3, 4. Similarly for y-coordinate, original rectangle includes y = -4 and y = 4, so strictly inside would be y from -3, -2, -1, 0, 1, 2, 3.

Therefore, the number of interior points would be the number of x-values times the number of y-values. Let me calculate that.

For x-values: from -4 to 4 inclusive. How many integers is that? Let me compute. From -4 to 4, that's 4 - (-4) + 1 = 9? Wait, no. Wait, the formula for the number of integers from a to b inclusive is b - a + 1. So here, if a is -4 and b is 4, then it's 4 - (-4) + 1 = 4 + 4 + 1 = 9. So 9 x-values. Similarly for y-values: from -3 to 3 inclusive. That would be 3 - (-3) + 1 = 3 + 3 + 1 = 7. So 7 y-values.

Therefore, the total number of interior points is 9 * 7 = 63.

Wait, but let me verify this with another approach to make sure.

Alternatively, the total number of integer points in the rectangle including the boundary is (number of x-values) * (number of y-values). For the original rectangle, x ranges from -5 to 5 inclusive. So that's 5 - (-5) + 1 = 11 x-values. Similarly, y ranges from -4 to 4 inclusive, which is 4 - (-4) + 1 = 9 y-values. So total points including boundary is 11 * 9 = 99.

But then, if we want strictly interior points, we need to subtract the boundary points. However, maybe it's easier to calculate directly as I did before. But let me check if my first approach was correct.

Alternatively, the number of interior points can be calculated by taking the total number of points in the rectangle and subtracting the points on the boundary. But that might be more complicated because of overlapping edges. However, maybe I can use inclusion-exclusion.

But perhaps my first approach is correct. Let me think again. If the rectangle is from x=-5 to x=5, then the interior points have x strictly between -5 and 5. Since x is integer, that means x can be -4, -3, ..., 4. Similarly for y. So that's 9 x-values and 7 y-values. So 9*7=63.

Alternatively, if I think of the rectangle as having width 10 (from -5 to 5 is 10 units) and height 8 (from -4 to 4 is 8 units). But the number of interior points would be (width - 2)*(height - 2). Wait, but that formula is for something else? Wait, no. Wait, if you have a grid, the number of interior points would be (number of points along x - 2)*(number of points along y - 2). Wait, maybe?

Wait, let me think. If the rectangle has width 10 units, but in terms of integer coordinates, the number of points along the x-axis is 11 (from -5 to 5 inclusive). So if we want interior points, we remove the two boundary points on each side. So, for x-axis, 11 - 2 = 9. Similarly for y-axis, original number of points is 9 (from -4 to 4 inclusive), so subtract 2, getting 7. Then 9*7=63. So that matches.

Alternatively, if I think of the formula for the number of interior points in a rectangle with integer coordinates. If the rectangle is defined by x from a to b inclusive and y from c to d inclusive, then the number of interior points is (b - a - 1)*(d - c - 1). Wait, let me check.

For example, if a rectangle from x=-5 to x=5 inclusive, then b - a - 1 = 5 - (-5) -1 = 10 -1 = 9. Similarly for y from -4 to 4, d - c -1 = 4 - (-4) -1 = 8 -1 = 7. So 9*7=63. So that formula works.

Therefore, the answer is 63.

But let me check with another method. Let me count the number of x-values and y-values again.

For x-coordinates strictly inside: original rectangle has x from -5 to 5. So the interior x-values are from -4 to 4. How many integers is that? Let me list them: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers.

Similarly for y-coordinates: original rectangle has y from -4 to 4. Interior y-values are from -3 to 3. That's -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

So 9*7=63.

Alternatively, if I think of the rectangle as centered at the origin, extending 5 units in x-direction and 4 units in y-direction. But the interior would be 4 units in x-direction (from -4 to 4) and 3 units in y-direction (from -3 to 3). Wait, but how does that translate to number of points?

Wait, in terms of distance, but in terms of coordinates, the number of points is as calculated before.

Alternatively, maybe I can think of the grid. For example, in the x-direction, between -5 and 5, there are 11 points. If I remove the two endpoints, I have 9 points. Similarly for y-direction, between -4 and 4, there are 9 points, remove two endpoints, 7 points.

So 9*7=63.

Alternatively, if I think of the formula for the number of lattice points inside a rectangle. The formula is (width - 2)*(height - 2). Wait, but here width is 10 (from -5 to 5 is 10 units), height is 8 (from -4 to 4 is 8 units). Then (10 - 2)*(8 - 2) = 8*6=48? Wait, that contradicts previous answer. Wait, so which is correct?

Wait, this is confusing. Wait, maybe the formula is different. Wait, if the rectangle has width W and height H, then the number of interior points is (W - 2)*(H - 2). But here, W is 10 units, H is 8 units. But then (10 - 2)*(8 - 2)= 8*6=48. But according to previous calculation, it's 63. So which is correct?

Wait, there's a confusion here between the number of units and the number of points. Let me clarify.

If the rectangle spans from x=-5 to x=5, that is 10 units in width. However, the number of integer points along the x-axis is 11 (from -5 to 5 inclusive). Similarly, the number of integer points along the y-axis is 9 (from -4 to 4 inclusive).

So if we want the number of interior points, we need to subtract 1 from each dimension? Wait, no. Wait, if you have N points along a line, then the number of interior points would be N - 2. For example, if you have 11 points along x-axis, then the interior points would be 11 - 2 = 9. Similarly for y-axis, 9 - 2 = 7. Therefore, total interior points is 9*7=63.

But if you think of the width as 10 units, then the number of interior points along x-axis would be 10 - 2 = 8? Wait, no. Wait, the number of interior points is not directly related to the width in units. It's related to the number of points.

So, for example, if you have a line segment from x=-5 to x=5, which is 10 units long, but there are 11 integer points on it. The interior points would be those not at the endpoints, so 11 - 2 = 9. Similarly, for the y-axis, from -4 to 4, which is 8 units long, with 9 integer points, so interior points are 9 - 2 = 7.

Therefore, the formula is (number of points along x - 2) * (number of points along y - 2). Which is (11 - 2)*(9 - 2) = 9*7=63.

But if you think of the width in units, then the number of interior points along x-axis would be (width - 2). Wait, but width is 10 units, so 10 - 2 = 8? But that would be different. So why is there a discrepancy?

Because the number of interior points along a line segment is not directly the width in units minus 2. It's the number of points minus 2. So if you have a line segment with N points, then the number of interior points is N - 2.

But N is equal to (length in units) + 1. For example, from -5 to 5 is 10 units, so N = 10 + 1 = 11. Therefore, interior points would be N - 2 = 9.

So if you have a width of W units, then the number of points along that width is W + 1. Therefore, the number of interior points is (W + 1) - 2 = W - 1. Wait, so if W is 10 units, then interior points would be 10 - 1 = 9? Wait, that seems conflicting with previous. Wait, no:

Wait, if W is the width in units, then number of points is W + 1. Then interior points is (W + 1) - 2 = W - 1. So for W = 10, interior points would be 10 - 1 = 9. Which matches. Similarly, for height H = 8 units, number of points is 8 + 1 = 9, so interior points is 9 - 2 = 7. Which is H - 1 = 8 - 1 = 7. So yes, that works.

Therefore, the formula for the number of interior points is (W - 1)*(H - 1). Wait, but W is the width in units, H is the height in units.

Wait, in our case, the rectangle has width from -5 to 5, which is 10 units, and height from -4 to 4, which is 8 units. Therefore, number of interior points is (10 - 1)*(8 - 1) = 9*7=63. Which matches our previous result.

So that formula works. Therefore, the answer is 63.

But let me check with a smaller rectangle to see if this formula works. Let's take a rectangle from (0,0) to (1,1). So the vertices are (0,0), (1,0), (1,1), (0,1). The interior points would be points strictly inside. Since the rectangle is 1x1, the only integer points are (0,0), (0,1), (1,0), (1,1), and (0.5, 0.5) but that's not integer. So there are no interior integer points.

Using the formula: width is 1 unit, height is 1 unit. So (1 - 1)*(1 - 1) = 0*0=0. Correct.

Another example: rectangle from (0,0) to (2,2). So vertices (0,0), (2,0), (2,2), (0,2). The interior points would be those with x from 1 to 1 (since strictly inside, so between 0 and 2, so x=1, y=1). So only (1,1). So number of interior points is 1.

Using the formula: width is 2 units, height is 2 units. (2 - 1)*(2 - 1)=1*1=1. Correct.

Another example: rectangle from (0,0) to (3,2). So width 3 units, height 2 units. The interior points would be x from 1 to 2 (since strictly inside between 0 and 3, so x=1,2) and y from 1 (since strictly inside between 0 and 2, so y=1). So number of interior points is 2*1=2.

Using formula: (3 - 1)*(2 - 1)=2*1=2. Correct.

So the formula seems to hold. Therefore, applying it to our original problem: width is 10 units, height is 8 units. So (10 - 1)*(8 - 1)=9*7=63.

Therefore, the answer is 63.

But just to be thorough, let me think again.

Original rectangle: x ranges from -5 to 5 inclusive. So the interior x-values are from -4 to 4 inclusive. How many numbers is that? Let me count. From -4 to 4, that's 4 - (-4) + 1 = 9 numbers. Similarly, y ranges from -4 to 4 inclusive, interior y-values are from -3 to 3 inclusive. That's 3 - (-3) + 1 = 7 numbers. Multiply them: 9*7=63.

Yes, same result.

Alternatively, if I think of the rectangle as having length 10 (from -5 to 5) and width 8 (from -4 to 4). The number of interior points is (length - 2)*(width - 2). Wait, but length is 10, so 10 - 2 = 8, width is 8, so 8 - 2 = 6. 8*6=48. Wait, that contradicts previous answer.

Wait, so why is there a discrepancy here?

Ah, because here, length and width are in units, but when we subtract 2, we are assuming that the number of interior points is (length - 2)*(width - 2). But earlier, we saw that the formula is (length - 1)*(width - 1).

Wait, so which is correct?

Wait, in the previous examples, when we had a rectangle from (0,0) to (1,1), length 1, width 1. Then (length - 1)*(width - 1) = 0*0=0, which is correct. But if we do (length - 2)*(width - 2) = (-1)*(-1)=1, which is wrong.

So clearly, the formula (length - 1)*(width - 1) is correct.

But why? Because length is the number of units, and the number of interior points is (number of units - 1) in each direction. Wait, but in the first example, with length 10 units, the number of interior points in x-direction is 9, which is 10 - 1. Similarly, for width 8 units, interior points is 7, which is 8 - 1.

Therefore, the formula is (length - 1)*(width - 1).

But why does this formula work? Because if you have a length of L units, then the number of integer points along that length is L + 1. Therefore, the number of interior points is (L + 1) - 2 = L - 1.

Therefore, the formula is (L - 1)*(W - 1).

Therefore, in our problem, L = 10, W = 8, so (10 - 1)*(8 - 1) = 9*7=63.

Therefore, the answer is 63.

But why did I think earlier that it was (length - 2)*(width - 2)? Maybe confusion with something else.

Alternatively, if you have a grid of points, the number of interior points is (number of points along x - 2)*(number of points along y - 2). Which is the same as ( (L + 1) - 2 )*( (W + 1) - 2 ) = (L - 1)*(W - 1).

Therefore, both approaches agree.

Therefore, the answer is 63.

But just to be 100% sure, let me count manually for a smaller rectangle.

Take the rectangle from (-1, -1) to (1, 1). So vertices at (-1,-1), (1,-1), (1,1), (-1,1). The interior points would be those strictly inside. So x from -0. something to 0. something, but integer coordinates. So x can be 0, and y can be 0. So only (0,0). So 1 point.

Using formula: length is 2 units (from -1 to 1), width is 2 units. So (2 - 1)*(2 - 1)=1*1=1. Correct.

Alternatively, number of points along x: from -1 to 1 inclusive is 3 points. Subtract 2, get 1. Similarly for y. 1*1=1. Correct.

Another example: rectangle from (-2, -2) to (2, 2). So interior points would be x from -1, 0, 1 and y from -1, 0, 1. So 3*3=9 points.

Using formula: length is 4 units, width is 4 units. (4 - 1)*(4 - 1)=3*3=9. Correct.

Alternatively, number of points along x: from -2 to 2 inclusive is 5 points. Subtract 2, get 3. Same for y. 3*3=9. Correct.

Therefore, formula holds.

Therefore, going back to original problem, answer is 63.

But just to make sure, let me think of another way.

Imagine that the rectangle is defined by -5 < x < 5 and -4 < y < 4. Since x and y are integers, x can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Similarly, y can be -3, -2, -1, 0, 1, 2, 3. That's 7 values.

So total number of points is 9*7=63.

Yes, that's straightforward.

Alternatively, if I think of the grid. For example, in the x-direction, from -5 to 5, there are 11 points. If I remove the two endpoints, I have 9. Similarly for y-direction, 9 points, remove two endpoints, 7.

Therefore, 9*7=63.

Therefore, I think the answer is 63.

But just to check, maybe I can think of the total number of points in the rectangle, which is 11*9=99. Then subtract the boundary points.

But how many boundary points are there?

The boundary consists of four sides. However, we have to be careful not to double count the corners.

Number of points on the bottom side: from (-5, -4) to (5, -4). That's 11 points.

Similarly, top side: from (-5, 4) to (5, 4). Also 11 points.

Left side: from (-5, -4) to (-5, 4). That's 9 points.

Right side: from (5, -4) to (5, 4). Also 9 points.

But the four corners are counted twice. Each corner is in two sides. So total boundary points: 11 + 11 + 9 + 9 - 4 = 30 - 4 = 26? Wait, 11 + 11 + 9 + 9 = 40. Then subtract 4 for the corners counted twice. So 40 - 4 = 36.

Wait, but let me check.

Total boundary points:

Bottom side: 11 points.

Top side: 11 points.

Left side: 9 points.

Right side: 9 points.

But the bottom side and top side include the left and right corners. Similarly, left and right sides include the top and bottom corners.

So total boundary points:

Bottom + top + left + right - 4 (for the four corners counted twice).

So 11 + 11 + 9 + 9 - 4 = 40 - 4 = 36.

Therefore, total boundary points are 36.

Therefore, total interior points would be total points (99) minus boundary points (36) = 63.

Yes! That matches.

So that's another way to get 63.

Therefore, this confirms that the answer is 63.

Therefore, I can be confident that the number of integer coordinate points strictly inside the rectangle is 63.

**Final Answer**
The number of interior points is \boxed{63}.

To determine how many integer-coordinate points lie **strictly inside** the rectangular region defined by the vertices $(5,4), (-5,4), (-5,-4), (5,-4)$, we proceed step by step.

---

### Step 1: Understand the Rectangle

The rectangle is aligned with the coordinate axes and spans:

- **X-direction**: from $-5$ to $5$
- **Y-direction**: from $-4$ to $4$

This means the rectangle includes all integer points $(x, y)$ such that:
$$
-5 \leq x \leq 5 \quad \text{and} \quad -4 \leq y \leq 4
$$

However, the question asks for **strictly interior** points, i.e., points that are **not on the boundary** of the rectangle.

---

### Step 2: Determine the Range of Interior Points

To be strictly inside the rectangle, the coordinates must satisfy:
$$
-5 < x < 5 \quad \text{and} \quad -4 < y < 4
$$

Since $x$ and $y$ must be **integers**, we find the integer values that satisfy these inequalities:

- For $x$: $-4, -3, -2, -1, 0, 1, 2, 3, 4$ → **9 values**
- For $y$: $-3, -2, -1, 0, 1, 2, 3$ → **7 values**

---

### Step 3: Compute the Total Number of Interior Points

Each interior point is a pair $(x, y)$ where $x$ is from the 9 values above and $y$ is from the 7 values above. So the total number of such points is:

$$
9 \times 7 = 63
$$

---

### Step 4: Verification via Total Points Minus Boundary Points

As a cross-check, we can compute the total number of integer points in the rectangle (including the boundary) and subtract the number of boundary points.

- **Total points in the rectangle**:
  $$
  (5 - (-5) + 1) \times (4 - (-4) + 1) = 11 \times 9 = 99
  $$

- **Boundary points**:
  - Bottom and top sides: $11$ points each → $22$
  - Left and right sides: $9$ points each → $18$
  - Subtract 4 for the 4 corners that are double-counted:
    $$
    22 + 18 - 4 = 36
    $$

- **Interior points**:
  $$
  99 - 36 = 63
  $$

This confirms our earlier result.

---

### Final Answer

$$
\boxed{63}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many integer coordinate points are strictly inside this rectangle with vertices at (5,4), (-5,4), (-5,-4), and (5,-4). Let me start by visualizing the rectangle. The rectangle spans from x = -5 to x = 5 and y = -4 to y = 4. But since the question is about points strictly inside, I need to exclude the boundaries.

First, let me recall that for a rectangle with sides parallel to the axes, the number of interior points can be calculated by considering the number of integer points along the x-axis and y-axis, then multiplying them. But since we need strictly interior points, we have to adjust for the boundaries.

Let me think. If the rectangle goes from x = -5 to x = 5, then the length along the x-axis is from -5 to 5. But since we want strictly interior points, we need to exclude the edges. So, for the x-coordinates, the interior points would be from -4 to 4, right? Because if you have a rectangle from -5 to 5, the interior would be points where x is greater than -5 and less than 5. But since we are dealing with integer coordinates, the integer points strictly inside would have x-values from -4 up to 4. Similarly for the y-coordinates, the original rectangle goes from -4 to 4, so the interior would be y-values from -3 up to 3.

Wait, let me check that. If the rectangle includes the boundary points, then the total number of integer points on the boundary would be... but actually, the question is about strictly interior points. So, for the x-coordinate, the original rectangle includes x = -5 and x = 5. So, to be strictly inside, x must be greater than -5 and less than 5. Since x is integer, that means x can be from -4, -3, -2, -1, 0, 1, 2, 3, 4. Similarly for y-coordinate, original rectangle includes y = -4 and y = 4, so strictly inside would be y from -3, -2, -1, 0, 1, 2, 3.

Therefore, the number of interior points would be the number of x-values times the number of y-values. Let me calculate that.

For x-values: from -4 to 4 inclusive. How many integers is that? Let me compute. From -4 to 4, that's 4 - (-4) + 1 = 9? Wait, no. Wait, the formula for the number of integers from a to b inclusive is b - a + 1. So here, if a is -4 and b is 4, then it's 4 - (-4) + 1 = 4 + 4 + 1 = 9. So 9 x-values. Similarly for y-values: from -3 to 3 inclusive. That would be 3 - (-3) + 1 = 3 + 3 + 1 = 7. So 7 y-values.

Therefore, the total number of interior points is 9 * 7 = 63.

Wait, but let me verify this with another approach to make sure.

Alternatively, the total number of integer points in the rectangle including the boundary is (number of x-values) * (number of y-values). For the original rectangle, x ranges from -5 to 5 inclusive. So that's 5 - (-5) + 1 = 11 x-values. Similarly, y ranges from -4 to 4 inclusive, which is 4 - (-4) + 1 = 9 y-values. So total points including boundary is 11 * 9 = 99.

But then, if we want strictly interior points, we need to subtract the boundary points. However, maybe it's easier to calculate directly as I did before. But let me check if my first approach was correct.

Alternatively, the number of interior points can be calculated by taking the total number of points in the rectangle and subtracting the points on the boundary. But that might be more complicated because of overlapping edges. However, maybe I can use inclusion-exclusion.

But perhaps my first approach is correct. Let me think again. If the rectangle is from x=-5 to x=5, then the interior points have x strictly between -5 and 5. Since x is integer, that means x can be -4, -3, ..., 4. Similarly for y. So that's 9 x-values and 7 y-values. So 9*7=63.

Alternatively, if I think of the rectangle as having width 10 (from -5 to 5 is 10 units) and height 8 (from -4 to 4 is 8 units). But the number of interior points would be (width - 2)*(height - 2). Wait, but that formula is for something else? Wait, no. Wait, if you have a grid, the number of interior points would be (number of points along x - 2)*(number of points along y - 2). Wait, maybe?

Wait, let me think. If the rectangle has width 10 units, but in terms of integer coordinates, the number of points along the x-axis is 11 (from -5 to 5 inclusive). So if we want interior points, we remove the two boundary points on each side. So, for x-axis, 11 - 2 = 9. Similarly for y-axis, original number of points is 9 (from -4 to 4 inclusive), so subtract 2, getting 7. Then 9*7=63. So that matches.

Alternatively, if I think of the formula for the number of interior points in a rectangle with integer coordinates. If the rectangle is defined by x from a to b inclusive and y from c to d inclusive, then the number of interior points is (b - a - 1)*(d - c - 1). Wait, let me check.

For example, if a rectangle from x=-5 to x=5 inclusive, then b - a - 1 = 5 - (-5) -1 = 10 -1 = 9. Similarly for y from -4 to 4, d - c -1 = 4 - (-4) -1 = 8 -1 = 7. So 9*7=63. So that formula works.

Therefore, the answer is 63.

But let me check with another method. Let me count the number of x-values and y-values again.

For x-coordinates strictly inside: original rectangle has x from -5 to 5. So the interior x-values are from -4 to 4. How many integers is that? Let me list them: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers.

Similarly for y-coordinates: original rectangle has y from -4 to 4. Interior y-values are from -3 to 3. That's -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

So 9*7=63.

Alternatively, if I think of the rectangle as centered at the origin, extending 5 units in x-direction and 4 units in y-direction. But the interior would be 4 units in x-direction (from -4 to 4) and 3 units in y-direction (from -3 to 3). Wait, but how does that translate to number of points?

Wait, in terms of distance, but in terms of coordinates, the number of points is as calculated before.

Alternatively, maybe I can think of the grid. For example, in the x-direction, between -5 and 5, there are 11 points. If I remove the two endpoints, I have 9 points. Similarly for y-direction, between -4 and 4, there are 9 points, remove two endpoints, 7 points.

So 9*7=63.

Alternatively, if I think of the formula for the number of lattice points inside a rectangle. The formula is (width - 2)*(height - 2). Wait, but here width is 10 (from -5 to 5 is 10 units), height is 8 (from -4 to 4 is 8 units). Then (10 - 2)*(8 - 2) = 8*6=48? Wait, that contradicts previous answer. Wait, so which is correct?

Wait, this is confusing. Wait, maybe the formula is different. Wait, if the rectangle has width W and height H, then the number of interior points is (W - 2)*(H - 2). But here, W is 10 units, H is 8 units. But then (10 - 2)*(8 - 2)= 8*6=48. But according to previous calculation, it's 63. So which is correct?

Wait, there's a confusion here between the number of units and the number of points. Let me clarify.

If the rectangle spans from x=-5 to x=5, that is 10 units in width. However, the number of integer points along the x-axis is 11 (from -5 to 5 inclusive). Similarly, the number of integer points along the y-axis is 9 (from -4 to 4 inclusive).

So if we want the number of interior points, we need to subtract 1 from each dimension? Wait, no. Wait, if you have N points along a line, then the number of interior points would be N - 2. For example, if you have 11 points along x-axis, then the interior points would be 11 - 2 = 9. Similarly for y-axis, 9 - 2 = 7. Therefore, total interior points is 9*7=63.

But if you think of the width as 10 units, then the number of interior points along x-axis would be 10 - 2 = 8? Wait, no. Wait, the number of interior points is not directly related to the width in units. It's related to the number of points.

So, for example, if you have a line segment from x=-5 to x=5, which is 10 units long, but there are 11 integer points on it. The interior points would be those not at the endpoints, so 11 - 2 = 9. Similarly, for the y-axis, from -4 to 4, which is 8 units long, with 9 integer points, so interior points are 9 - 2 = 7.

Therefore, the formula is (number of points along x - 2) * (number of points along y - 2). Which is (11 - 2)*(9 - 2) = 9*7=63.

But if you think of the width in units, then the number of interior points along x-axis would be (width - 2). Wait, but width is 10 units, so 10 - 2 = 8? But that would be different. So why is there a discrepancy?

Because the number of interior points along a line segment is not directly the width in units minus 2. It's the number of points minus 2. So if you have a line segment with N points, then the number of interior points is N - 2.

But N is equal to (length in units) + 1. For example, from -5 to 5 is 10 units, so N = 10 + 1 = 11. Therefore, interior points would be N - 2 = 9.

So if you have a width of W units, then the number of points along that width is W + 1. Therefore, the number of interior points is (W + 1) - 2 = W - 1. Wait, so if W is 10 units, then interior points would be 10 - 1 = 9? Wait, that seems conflicting with previous. Wait, no:

Wait, if W is the width in units, then number of points is W + 1. Then interior points is (W + 1) - 2 = W - 1. So for W = 10, interior points would be 10 - 1 = 9. Which matches. Similarly, for height H = 8 units, number of points is 8 + 1 = 9, so interior points is 9 - 2 = 7. Which is H - 1 = 8 - 1 = 7. So yes, that works.

Therefore, the formula for the number of interior points is (W - 1)*(H - 1). Wait, but W is the width in units, H is the height in units.

Wait, in our case, the rectangle has width from -5 to 5, which is 10 units, and height from -4 to 4, which is 8 units. Therefore, number of interior points is (10 - 1)*(8 - 1) = 9*7=63. Which matches our previous result.

So that formula works. Therefore, the answer is 63.

But let me check with a smaller rectangle to see if this formula works. Let's take a rectangle from (0,0) to (1,1). So the vertices are (0,0), (1,0), (1,1), (0,1). The interior points would be points strictly inside. Since the rectangle is 1x1, the only integer points are (0,0), (0,1), (1,0), (1,1), and (0.5, 0.5) but that's not integer. So there are no interior integer points.

Using the formula: width is 1 unit, height is 1 unit. So (1 - 1)*(1 - 1) = 0*0=0. Correct.

Another example: rectangle from (0,0) to (2,2). So vertices (0,0), (2,0), (2,2), (0,2). The interior points would be those with x from 1 to 1 (since strictly inside, so between 0 and 2, so x=1, y=1). So only (1,1). So number of interior points is 1.

Using the formula: width is 2 units, height is 2 units. (2 - 1)*(2 - 1)=1*1=1. Correct.

Another example: rectangle from (0,0) to (3,2). So width 3 units, height 2 units. The interior points would be x from 1 to 2 (since strictly inside between 0 and 3, so x=1,2) and y from 1 (since strictly inside between 0 and 2, so y=1). So number of interior points is 2*1=2.

Using formula: (3 - 1)*(2 - 1)=2*1=2. Correct.

So the formula seems to hold. Therefore, applying it to our original problem: width is 10 units, height is 8 units. So (10 - 1)*(8 - 1)=9*7=63.

Therefore, the answer is 63.

But just to be thorough, let me think again.

Original rectangle: x ranges from -5 to 5 inclusive. So the interior x-values are from -4 to 4 inclusive. How many numbers is that? Let me count. From -4 to 4, that's 4 - (-4) + 1 = 9 numbers. Similarly, y ranges from -4 to 4 inclusive, interior y-values are from -3 to 3 inclusive. That's 3 - (-3) + 1 = 7 numbers. Multiply them: 9*7=63.

Yes, same result.

Alternatively, if I think of the rectangle as having length 10 (from -5 to 5) and width 8 (from -4 to 4). The number of interior points is (length - 2)*(width - 2). Wait, but length is 10, so 10 - 2 = 8, width is 8, so 8 - 2 = 6. 8*6=48. Wait, that contradicts previous answer.

Wait, so why is there a discrepancy here?

Ah, because here, length and width are in units, but when we subtract 2, we are assuming that the number of interior points is (length - 2)*(width - 2). But earlier, we saw that the formula is (length - 1)*(width - 1).

Wait, so which is correct?

Wait, in the previous examples, when we had a rectangle from (0,0) to (1,1), length 1, width 1. Then (length - 1)*(width - 1) = 0*0=0, which is correct. But if we do (length - 2)*(width - 2) = (-1)*(-1)=1, which is wrong.

So clearly, the formula (length - 1)*(width - 1) is correct.

But why? Because length is the number of units, and the number of interior points is (number of units - 1) in each direction. Wait, but in the first example, with length 10 units, the number of interior points in x-direction is 9, which is 10 - 1. Similarly, for width 8 units, interior points is 7, which is 8 - 1.

Therefore, the formula is (length - 1)*(width - 1).

But why does this formula work? Because if you have a length of L units, then the number of integer points along that length is L + 1. Therefore, the number of interior points is (L + 1) - 2 = L - 1.

Therefore, the formula is (L - 1)*(W - 1).

Therefore, in our problem, L = 10, W = 8, so (10 - 1)*(8 - 1) = 9*7=63.

Therefore, the answer is 63.

But why did I think earlier that it was (length - 2)*(width - 2)? Maybe confusion with something else.

Alternatively, if you have a grid of points, the number of interior points is (number of points along x - 2)*(number of points along y - 2). Which is the same as ( (L + 1) - 2 )*( (W + 1) - 2 ) = (L - 1)*(W - 1).

Therefore, both approaches agree.

Therefore, the answer is 63.

But just to be 100% sure, let me count manually for a smaller rectangle.

Take the rectangle from (-1, -1) to (1, 1). So vertices at (-1,-1), (1,-1), (1,1), (-1,1). The interior points would be those strictly inside. So x from -0. something to 0. something, but integer coordinates. So x can be 0, and y can be 0. So only (0,0). So 1 point.

Using formula: length is 2 units (from -1 to 1), width is 2 units. So (2 - 1)*(2 - 1)=1*1=1. Correct.

Alternatively, number of points along x: from -1 to 1 inclusive is 3 points. Subtract 2, get 1. Similarly for y. 1*1=1. Correct.

Another example: rectangle from (-2, -2) to (2, 2). So interior points would be x from -1, 0, 1 and y from -1, 0, 1. So 3*3=9 points.

Using formula: length is 4 units, width is 4 units. (4 - 1)*(4 - 1)=3*3=9. Correct.

Alternatively, number of points along x: from -2 to 2 inclusive is 5 points. Subtract 2, get 3. Same for y. 3*3=9. Correct.

Therefore, formula holds.

Therefore, going back to original problem, answer is 63.

But just to make sure, let me think of another way.

Imagine that the rectangle is defined by -5 < x < 5 and -4 < y < 4. Since x and y are integers, x can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Similarly, y can be -3, -2, -1, 0, 1, 2, 3. That's 7 values.

So total number of points is 9*7=63.

Yes, that's straightforward.

Alternatively, if I think of the grid. For example, in the x-direction, from -5 to 5, there are 11 points. If I remove the two endpoints, I have 9. Similarly for y-direction, 9 points, remove two endpoints, 7.

Therefore, 9*7=63.

Therefore, I think the answer is 63.

But just to check, maybe I can think of the total number of points in the rectangle, which is 11*9=99. Then subtract the boundary points.

But how many boundary points are there?

The boundary consists of four sides. However, we have to be careful not to double count the corners.

Number of points on the bottom side: from (-5, -4) to (5, -4). That's 11 points.

Similarly, top side: from (-5, 4) to (5, 4). Also 11 points.

Left side: from (-5, -4) to (-5, 4). That's 9 points.

Right side: from (5, -4) to (5, 4). Also 9 points.

But the four corners are counted twice. Each corner is in two sides. So total boundary points: 11 + 11 + 9 + 9 - 4 = 30 - 4 = 26? Wait, 11 + 11 + 9 + 9 = 40. Then subtract 4 for the corners counted twice. So 40 - 4 = 36.

Wait, but let me check.

Total boundary points:

Bottom side: 11 points.

Top side: 11 points.

Left side: 9 points.

Right side: 9 points.

But the bottom side and top side include the left and right corners. Similarly, left and right sides include the top and bottom corners.

So total boundary points:

Bottom + top + left + right - 4 (for the four corners counted twice).

So 11 + 11 + 9 + 9 - 4 = 40 - 4 = 36.

Therefore, total boundary points are 36.

Therefore, total interior points would be total points (99) minus boundary points (36) = 63.

Yes! That matches.

So that's another way to get 63.

Therefore, this confirms that the answer is 63.

Therefore, I can be confident that the number of integer coordinate points strictly inside the rectangle is 63.

**Final Answer**
The number of interior points is \boxed{63}.

To determine how many integer-coordinate points lie **strictly inside** the rectangular region defined by the vertices $(5,4), (-5,4), (-5,-4), (5,-4)$, we proceed step by step.

---

### Step 1: Understand the Rectangle

The rectangle is aligned with the coordinate axes and spans:

- **X-direction**: from $-5$ to $5$
- **Y-direction**: from $-4$ to $4$

This means the rectangle includes all integer points $(x, y)$ such that:
$$
-5 \leq x \leq 5 \quad \text{and} \quad -4 \leq y \leq 4
$$

However, the question asks for **strictly interior** points, i.e., points that are **not on the boundary** of the rectangle.

---

### Step 2: Determine the Range of Interior Points

To be strictly inside the rectangle, the coordinates must satisfy:
$$
-5 < x < 5 \quad \text{and} \quad -4 < y < 4
$$

Since $x$ and $y$ must be **integers**, we find the integer values that satisfy these inequalities:

- For $x$: $-4, -3, -2, -1, 0, 1, 2, 3, 4$ → **9 values**
- For $y$: $-3, -2, -1, 0, 1, 2, 3$ → **7 values**

---

### Step 3: Compute the Total Number of Interior Points

Each interior point is a pair $(x, y)$ where $x$ is from the 9 values above and $y$ is from the 7 values above. So the total number of such points is:

$$
9 \times 7 = 63
$$

---

### Step 4: Verification via Total Points Minus Boundary Points

As a cross-check, we can compute the total number of integer points in the rectangle (including the boundary) and subtract the number of boundary points.

- **Total points in the rectangle**:
  $$
  (5 - (-5) + 1) \times (4 - (-4) + 1) = 11 \times 9 = 99
  $$

- **Boundary points**:
  - Bottom and top sides: $11$ points each → $22$
  - Left and right sides: $9$ points each → $18$
  - Subtract 4 for the 4 corners that are double-counted:
    $$
    22 + 18 - 4 = 36
    $$

- **Interior points**:
  $$
  99 - 36 = 63
  $$

This confirms our earlier result.

---

### Final Answer

$$
\boxed{63}
$$

Solution:
The rectangular region is 10 units by 8 units, resulting in an 8 by 6 rectangular region in the interior, which forms a 9-by-7 array of lattice points. That's $\boxed{63}$ points with integer coordinates, as shown in the figure. [asy]
import olympiad; size(150); defaultpen(linewidth(0.8));
add(grid(10,8));
draw((1,1)--(9,1)--(9,7)--(1,7)--cycle,linewidth(1.2));
[/asy] Note: We are counting points, not squares. It is a common mistake to count the interior squares, for 48, instead of interior lattice points, which gives the correct answer of 63.


Explanation:
Okay, so I need to find the value of $ x_{531} + x_{753} + x_{975} $ given this recurrence relation. Let me start by understanding the problem step by step.

First, the sequence is defined with the first four terms:

- $ x_1 = 211 $
- $ x_2 = 375 $
- $ x_3 = 420 $
- $ x_4 = 523 $

And for $ n \geq 5 $, each term is defined by the recurrence relation:

$ x_n = x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} $

Hmm, so each term depends on the previous four terms. This seems like a linear recurrence relation. Maybe I can find a pattern or periodicity in the sequence? Because if the sequence is periodic, then I can find the period and compute the required terms modulo that period.

Let me try to compute the first few terms to see if there's a pattern or repetition.

Starting with the given terms:

$ x_1 = 211 $

$ x_2 = 375 $

$ x_3 = 420 $

$ x_4 = 523 $

Now compute $ x_5 $:

$ x_5 = x_4 - x_3 + x_2 - x_1 = 523 - 420 + 375 - 211 $

Let me calculate that step by step:

523 - 420 = 103

103 + 375 = 478

478 - 211 = 267

So $ x_5 = 267 $

Now $ x_6 = x_5 - x_4 + x_3 - x_2 $

Plugging in the numbers:

267 - 523 + 420 - 375

Compute step by step:

267 - 523 = -256

-256 + 420 = 164

164 - 375 = -211

So $ x_6 = -211 $

Next, $ x_7 = x_6 - x_5 + x_4 - x_3 $

-211 - 267 + 523 - 420

Calculating:

-211 - 267 = -478

-478 + 523 = 45

45 - 420 = -375

So $ x_7 = -375 $

Then $ x_8 = x_7 - x_6 + x_5 - x_4 $

-375 - (-211) + 267 - 523

Simplify:

-375 + 211 = -164

-164 + 267 = 103

103 - 523 = -420

So $ x_8 = -420 $

Next, $ x_9 = x_8 - x_7 + x_6 - x_5 $

-420 - (-375) + (-211) - 267

Calculating:

-420 + 375 = -45

-45 - 211 = -256

-256 - 267 = -523

So $ x_9 = -523 $

Now $ x_{10} = x_9 - x_8 + x_7 - x_6 $

-523 - (-420) + (-375) - (-211)

Simplify:

-523 + 420 = -103

-103 - 375 = -478

-478 + 211 = -267

So $ x_{10} = -267 $

Moving on to $ x_{11} = x_{10} - x_9 + x_8 - x_7 $

-267 - (-523) + (-420) - (-375)

Calculating:

-267 + 523 = 256

256 - 420 = -164

-164 + 375 = 211

So $ x_{11} = 211 $

Wait a second! $ x_{11} = 211 $, which is the same as $ x_1 $. Let me check if the following terms also repeat.

Compute $ x_{12} = x_{11} - x_{10} + x_9 - x_8 $

211 - (-267) + (-523) - (-420)

Simplify:

211 + 267 = 478

478 - 523 = -45

-45 + 420 = 375

So $ x_{12} = 375 $, which is $ x_2 $.

Then $ x_{13} = x_{12} - x_{11} + x_{10} - x_9 $

375 - 211 + (-267) - (-523)

Calculating:

375 - 211 = 164

164 - 267 = -103

-103 + 523 = 420

So $ x_{13} = 420 $, which is $ x_3 $.

Next, $ x_{14} = x_{13} - x_{12} + x_{11} - x_{10} $

420 - 375 + 211 - (-267)

Calculating:

420 - 375 = 45

45 + 211 = 256

256 + 267 = 523

So $ x_{14} = 523 $, which is $ x_4 $.

Then $ x_{15} = x_{14} - x_{13} + x_{12} - x_{11} $

523 - 420 + 375 - 211

Wait, that's exactly the same as $ x_5 $, which was 267. So $ x_{15} = 267 $, same as $ x_5 $.

So it seems like the sequence is periodic with period 10. Let me check:

Original terms:

x1=211, x2=375, x3=420, x4=523, x5=267, x6=-211, x7=-375, x8=-420, x9=-523, x10=-267, x11=211, x12=375,...

Yes! So starting from x1, the sequence repeats every 10 terms. Therefore, the period is 10.

So the sequence is periodic with period 10. Therefore, to find x_n, we can compute n modulo 10, and then find the corresponding term in the first 10 terms.

But let me confirm this periodicity. Let me check a few more terms to be sure.

x11=211 (same as x1)

x12=375 (same as x2)

x13=420 (same as x3)

x14=523 (same as x4)

x15=267 (same as x5)

x16= x15 - x14 + x13 - x12 = 267 - 523 + 420 - 375 = same as x6: -211

Yes, same as x6. So the pattern continues. Therefore, the period is indeed 10.

Therefore, the sequence repeats every 10 terms. Therefore, for any n, x_n = x_{n mod 10}, but we have to be careful with the modulus. Let me check:

For example, n=11: 11 mod 10 = 1, so x11 = x1. Similarly, n=12: 12 mod 10 = 2, so x12 = x2. However, note that when n is a multiple of 10, say n=10, 10 mod 10 = 0. But in our case, the original terms start at n=1, so perhaps we need to adjust the modulus accordingly.

Wait, let me think. If the period is 10, then x_{k} = x_{k + 10} for all k ≥ 1. Therefore, to find x_n, we can compute n modulo 10, but if the remainder is 0, then it corresponds to the 10th term. For example:

If n=10, then 10 mod 10 = 0, but we need to map that to x10. So perhaps we can define:

Let m = n mod 10. If m = 0, then the term is x10; otherwise, it's x_m.

Alternatively, since the period is 10, the sequence is:

Term number: 1 2 3 4 5 6 7 8 9 10 11 12 ...

Value:        211, 375, 420, 523, 267, -211, -375, -420, -523, -267, 211, 375,...

So for term number n, if we compute (n - 1) mod 10, then:

For n=1: (1-1)=0 mod10=0 → index 0? But our first term is index 1. Maybe another way.

Alternatively, since the period is 10, then x_{n} = x_{n - 10} for n ≥ 11. Therefore, to find x_n, we can subtract 10 until we get into the first 10 terms.

Therefore, for any n, x_n = x_{n mod 10}, but if n mod 10 is 0, then it's x_{10}.

Alternatively, if we think of the sequence as starting at n=1, then the position within the period can be calculated as:

Let’s define k = n - 1. Then k mod 10 gives 0 to 9, which corresponds to the original terms 1 to 10. Therefore, x_n = x_{(k mod 10) + 1} where k = n - 1.

So for example, n=1: k=0, 0 mod10=0, so term is x_{0+1}=x1. Similarly, n=10: k=9, 9 mod10=9, term is x_{9+1}=x10. n=11: k=10, 10 mod10=0, term is x_{0+1}=x1. Perfect.

Therefore, in general, x_n = x_{(n - 1) mod 10 + 1}

Therefore, to compute x_n, we can compute (n - 1) mod 10, add 1, and get the position in the first 10 terms.

Therefore, for example, x_{531}:

Compute (531 - 1) mod 10 = 530 mod 10 = 0. Then add 1: 0 + 1 = 1. Therefore, x_{531} = x1 = 211.

Similarly, x_{753}:

(753 - 1) mod 10 = 752 mod 10 = 2. Add 1: 3. Therefore, x_{753} = x3 = 420.

Wait, hold on. Wait, let me check:

Wait, if (n - 1) mod 10 gives 0 to 9, then adding 1 gives 1 to 10. Therefore, if (n - 1) mod 10 = 0, then x_n = x1; if it's 1, x2; ... if it's 9, x10.

Therefore, for n=531:

(531 - 1) = 530. 530 divided by 10 is 53, remainder 0. Therefore, (n - 1) mod 10 = 0. Therefore, x_{531} = x1 = 211.

Similarly, n=753:

(753 - 1) = 752. 752 divided by 10 is 75*10=750, remainder 2. Therefore, (n - 1) mod 10 = 2. Therefore, x_{753} = x_{2 + 1} = x3? Wait, no:

Wait, if (n - 1) mod 10 = 2, then adding 1 gives 3? Wait, no:

Wait, the formula is x_n = x_{(n - 1) mod 10 + 1}

So if (n - 1) mod 10 = 0, then x_n = x_{0 + 1} = x1

If (n - 1) mod 10 = 1, then x_n = x_{1 + 1} = x2

...

If (n - 1) mod 10 = 9, then x_n = x_{9 + 1} = x10

Therefore, for n=753:

(n - 1) mod 10 = 752 mod 10 = 2. Therefore, x_{753} = x_{2 + 1} = x3? Wait, no:

Wait, (n - 1) mod 10 = 2, so the position is 2 + 1? Wait, no. Wait, the formula is:

Let me think again. Let me take n=1:

(n - 1) = 0, mod10=0, so position is 0 + 1 = 1. Correct.

n=2: (2 - 1)=1 mod10=1, position=1 + 1=2. Correct.

Wait, so the formula is x_n = x_{(n - 1) mod 10 + 1}

Therefore, if (n - 1) mod 10 = k, then x_n = x_{k + 1}

Therefore, for n=753:

k = (753 - 1) mod 10 = 752 mod 10 = 2. Therefore, x_{753} = x_{2 + 1} = x3. Which is 420. Correct.

Similarly, for n=975:

(975 - 1) = 974. 974 mod 10 = 4. Therefore, x_{975} = x_{4 + 1} = x5. Which is 267.

Wait, so then x_{531} + x_{753} + x_{975} = 211 + 420 + 267.

Let me compute that:

211 + 420 = 631; 631 + 267 = 898.

But wait, let me check again if my reasoning is correct. Because if the period is 10, then every 10 terms it repeats. Therefore, the value of x_n depends on n mod 10. However, I need to make sure that my calculation for each term is correct.

Let me check for n=10:

(n - 1) = 9, mod10=9, so x_{10} = x_{9 + 1} = x10. Correct.

n=11: (11 - 1)=10 mod10=0, so x_{11}=x_{0 + 1}=x1. Correct.

Therefore, the formula is correct.

So for n=531:

531 divided by 10 is 53*10 + 1. Therefore, 531 = 53*10 + 1. Therefore, the remainder is 1. Therefore, since the period is 10, x_{531} = x1 = 211. Correct.

Similarly, n=753:

Divide 753 by 10: 75*10 = 750, 753 - 750 = 3. Therefore, remainder is 3. Therefore, x_{753} = x3 = 420. Wait, but according to previous calculation, (n - 1) mod10 = 2. Wait, there's a discrepancy here. Wait, which is correct?

Wait, if n=753, then n mod10 is 3. But according to the formula, (n - 1) mod10 = 752 mod10=2. Therefore, x_{753} = x_{2 + 1} = x3. Which is same as n mod10=3, so x_{n} = x_{n mod10} if n mod10 ≠0, else x_{10}. Wait, but here n mod10=3, so x_{753}=x3. Which is same as the formula. So both ways, if I think of n mod10, then:

If n mod10 = 0, then x_n = x10, else x_n = x_{n mod10}

But in our case, for n=753, n mod10=3, so x_{753}=x3. Which is correct. So both methods are equivalent.

Therefore, perhaps another way to think is:

If the period is 10, then x_n = x_{n mod10} if n mod10 ≠0, else x_{10}

But in our case, since the first term is x1, so for n=10, which is 10 mod10=0, we have x_{10}.

Therefore, for n=531:

531 divided by 10 is 53*10 + 1, so remainder 1. Therefore, x_{531}=x1=211.

For n=753:

753 divided by 10 is 75*10 + 3, remainder 3. Therefore, x_{753}=x3=420.

For n=975:

975 divided by 10 is 97*10 + 5, remainder 5. Therefore, x_{975}=x5=267.

Therefore, adding them up: 211 + 420 + 267.

Let me compute that again:

211 + 420 = 631

631 + 267:

631 + 200 = 831

831 + 67 = 898

So total is 898.

But wait, let me check if my calculation of x5 is correct. Earlier, I calculated x5 as 267. Let me check again:

x5 = x4 - x3 + x2 - x1 = 523 - 420 + 375 - 211.

Compute step by step:

523 - 420 = 103

103 + 375 = 478

478 - 211 = 267. Yes, correct.

So x5=267. Therefore, x_{975}=x5=267. So the sum is 211 + 420 + 267 = 898.

But wait, let me check if there's a mistake in the periodicity. Because when I calculated up to x10, then x11=x1, so the period is 10. But let me check x10 and x_{20} to be sure.

Earlier, x10=-267. Then x11=211, x12=375,..., x20 would be same as x10, which is -267. Let me check x20:

But maybe it's tedious, but since the recurrence is linear and we saw that after 10 terms it repeats, and the recurrence is of order 4, but with period 10, which is longer than the order. However, since we saw that the sequence repeats after 10 terms, and the calculation for x11 to x20 would follow the same pattern, so I think the periodicity is correct.

Therefore, the answer should be 898. However, let me check once again for n=531, 753, 975.

Alternatively, maybe I can check with another approach. Let me compute n mod 10 for each term:

For 531:

Divide 531 by 10: 53*10=530, 531-530=1. So remainder 1. Therefore, x_{531}=x1=211.

For 753:

753 divided by 10: 75*10=750, 753-750=3. Remainder 3. Therefore, x_{753}=x3=420.

For 975:

975 divided by 10: 97*10=970, 975-970=5. Remainder 5. Therefore, x_{975}=x5=267.

Therefore, sum is 211 + 420 + 267 = 898. So that seems correct.

But let me check if there's a possibility that the period is different? For example, maybe the period is 20 or something else? Wait, but we saw that x11=x1, x12=x2,..., so the period is 10. So unless there is a mistake in my calculation of the terms, which I checked up to x14, which repeats x4, so seems correct.

Alternatively, maybe I can check the recurrence relation for the periodicity. Suppose that the sequence is periodic with period 10, then for all n ≥ 1, x_{n+10} = x_n. Let me check for n=1:

x_{11} = x1 = 211. Correct.

Similarly, x_{12}=x2=375. Correct.

So if this holds for all n, then the period is indeed 10. Therefore, the answer is 898.

But let me check another term. For example, x_{10} = -267. Then x_{20} should be equal to x_{10} = -267. Let me compute x_{10} = -267, then x_{11}=211, x_{12}=375, x_{13}=420, x_{14}=523, x_{15}=267, x_{16}=-211, x_{17}=-375, x_{18}=-420, x_{19}=-523, x_{20}= -267. Yes, same as x_{10}. Therefore, the period is indeed 10. Therefore, the conclusion is correct.

Therefore, the answer is 211 + 420 + 267 = 898.

But wait, let me check once again the calculation of the sum:

211 + 420 = 631. Then 631 + 267:

Let me compute 631 + 200 = 831, then 831 + 67 = 898. Yes.

Alternatively, 211 + 267 = 478, then 478 + 420 = 898. Same result.

Therefore, I think the answer is 898.

But just to be thorough, let me check if there's a possibility that the problem is in the indexing. For example, sometimes sequences start at n=0, but here it's given as n=1, so I think the calculation is correct.

Alternatively, maybe I made a mistake in the initial terms? Let me check again:

x1=211, x2=375, x3=420, x4=523.

x5 = x4 - x3 + x2 - x1 = 523 - 420 + 375 - 211 = 523 - 420 is 103, 103 + 375 is 478, 478 - 211 is 267. Correct.

x6 = x5 - x4 + x3 - x2 = 267 - 523 + 420 - 375. 267 - 523 = -256, -256 + 420 = 164, 164 - 375 = -211. Correct.

x7 = x6 - x5 + x4 - x3 = -211 - 267 + 523 - 420. -211 -267 = -478, -478 + 523 = 45, 45 - 420 = -375. Correct.

x8 = x7 - x6 + x5 - x4 = -375 - (-211) + 267 - 523 = -375 + 211 = -164, -164 + 267 = 103, 103 - 523 = -420. Correct.

x9 = x8 - x7 + x6 - x5 = -420 - (-375) + (-211) - 267 = -420 + 375 = -45, -45 -211 = -256, -256 -267 = -523. Correct.

x10 = x9 - x8 + x7 - x6 = -523 - (-420) + (-375) - (-211) = -523 + 420 = -103, -103 - 375 = -478, -478 + 211 = -267. Correct.

x11 = x10 - x9 + x8 - x7 = -267 - (-523) + (-420) - (-375) = -267 + 523 = 256, 256 - 420 = -164, -164 + 375 = 211. Correct. So x11 = x1. All steps are correct.

Therefore, the periodicity is confirmed. Therefore, the answer is 898.

But just to be 100% sure, let me check another term. Let's compute x_{15} which should be equal to x5=267.

x11=211, x12=375, x13=420, x14=523, x15 = x14 - x13 + x12 - x11 = 523 - 420 + 375 - 211 = same as x5, which is 267. Correct.

Similarly, x_{16} = x15 - x14 + x13 - x12 = 267 - 523 + 420 - 375 = same as x6=-211. Correct.

Therefore, the periodicity is confirmed. Therefore, the answer is 898.

But wait, the problem says "find the value of $ x_{531} + x_{753} + x_{975} $." So according to my calculation, it's 211 + 420 + 267 = 898. However, let me check once again the values for each term:

For x_{531}:

Since 531 divided by 10 is 53*10 + 1, so remainder 1. Therefore, x_{531} = x1 = 211. Correct.

For x_{753}:

753 divided by 10 is 75*10 + 3, remainder 3. Therefore, x_{753} = x3 = 420. Correct.

For x_{975}:

975 divided by 10 is 97*10 + 5, remainder 5. Therefore, x_{975} = x5 = 267. Correct.

Therefore, adding them up: 211 + 420 + 267 = 898. So I think that's the answer.

But just to be thorough, let me check if there's an alternative way to compute this, maybe using the recurrence relation properties or matrix exponentiation or something else, but since the period is 10, it's straightforward.

Alternatively, maybe there's a mistake in assuming the period is 10? Let me check for n=10, x_{10}=-267, and then x_{10 + 10}=x_{20}=-267. Let me compute x_{15}=267, x_{16}=-211, x_{17}=-375, x_{18}=-420, x_{19}=-523, x_{20}= -267. Yes, same as x_{10}. Therefore, the period is indeed 10.

Therefore, I think my answer is correct. Therefore, the value is 898.

**Final Answer**
\boxed{898}

To solve the problem, we are given a recurrence relation:

$$
\begin{align*}
x_1 &= 211, \\
x_2 &= 375, \\
x_3 &= 420, \\
x_4 &= 523, \\
x_n &= x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} \quad \text{for } n \geq 5.
\end{align*}
$$

We are asked to compute:

$$
x_{531} + x_{753} + x_{975}
$$

---

### Step 1: Identify the Pattern

We begin by computing the first few terms of the sequence to look for a pattern or periodicity.

$$
\begin{align*}
x_1 &= 211 \\
x_2 &= 375 \\
x_3 &= 420 \\
x_4 &= 523 \\
x_5 &= x_4 - x_3 + x_2 - x_1 = 523 - 420 + 375 - 211 = 267 \\
x_6 &= x_5 - x_4 + x_3 - x_2 = 267 - 523 + 420 - 375 = -211 \\
x_7 &= x_6 - x_5 + x_4 - x_3 = -211 - 267 + 523 - 420 = -375 \\
x_8 &= x_7 - x_6 + x_5 - x_4 = -375 - (-211) + 267 - 523 = -420 \\
x_9 &= x_8 - x_7 + x_6 - x_5 = -420 - (-375) + (-211) - 267 = -523 \\
x_{10} &= x_9 - x_8 + x_7 - x_6 = -523 - (-420) + (-375) - (-211) = -267 \\
x_{11} &= x_{10} - x_9 + x_8 - x_7 = -267 - (-523) + (-420) - (-375) = 211 \\
x_{12} &= x_{11} - x_{10} + x_9 - x_8 = 211 - (-267) + (-523) - (-420) = 375 \\
x_{13} &= x_{12} - x_{11} + x_{10} - x_9 = 375 - 211 + (-267) - (-523) = 420 \\
x_{14} &= x_{13} - x_{12} + x_{11} - x_{10} = 420 - 375 + 211 - (-267) = 523 \\
x_{15} &= x_{14} - x_{13} + x_{12} - x_{11} = 523 - 420 + 375 - 211 = 267 \\
\end{align*}
$$

We observe that:

$$
x_{11} = x_1, \quad x_{12} = x_2, \quad x_{13} = x_3, \quad x_{14} = x_4, \quad x_{15} = x_5
$$

This confirms that the sequence is **periodic with period 10**.

---

### Step 2: Use the Periodicity to Compute the Required Terms

Since the sequence is periodic with period 10, we can compute any term $ x_n $ by finding the position of $ n $ within the first 10 terms.

To do this, we compute:

$$
n \mod 10
$$

- If the remainder is 0, then $ x_n = x_{10} $
- Otherwise, $ x_n = x_{\text{remainder}} $

Let’s compute each term:

- $ x_{531} $: $ 531 \mod 10 = 1 \Rightarrow x_{531} = x_1 = 211 $
- $ x_{753} $: $ 753 \mod 10 = 3 \Rightarrow x_{753} = x_3 = 420 $
- $ x_{975} $: $ 975 \mod 10 = 5 \Rightarrow x_{975} = x_5 = 267 $

---

### Step 3: Compute the Final Sum

$$
x_{531} + x_{753} + x_{975} = 211 + 420 + 267 = \boxed{898}
$$

---

### Final Answer

$$
\boxed{898}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of $ x_{531} + x_{753} + x_{975} $ given this recurrence relation. Let me start by understanding the problem step by step.

First, the sequence is defined with the first four terms:

- $ x_1 = 211 $
- $ x_2 = 375 $
- $ x_3 = 420 $
- $ x_4 = 523 $

And for $ n \geq 5 $, each term is defined by the recurrence relation:

$ x_n = x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} $

Hmm, so each term depends on the previous four terms. This seems like a linear recurrence relation. Maybe I can find a pattern or periodicity in the sequence? Because if the sequence is periodic, then I can find the period and compute the required terms modulo that period.

Let me try to compute the first few terms to see if there's a pattern or repetition.

Starting with the given terms:

$ x_1 = 211 $

$ x_2 = 375 $

$ x_3 = 420 $

$ x_4 = 523 $

Now compute $ x_5 $:

$ x_5 = x_4 - x_3 + x_2 - x_1 = 523 - 420 + 375 - 211 $

Let me calculate that step by step:

523 - 420 = 103

103 + 375 = 478

478 - 211 = 267

So $ x_5 = 267 $

Now $ x_6 = x_5 - x_4 + x_3 - x_2 $

Plugging in the numbers:

267 - 523 + 420 - 375

Compute step by step:

267 - 523 = -256

-256 + 420 = 164

164 - 375 = -211

So $ x_6 = -211 $

Next, $ x_7 = x_6 - x_5 + x_4 - x_3 $

-211 - 267 + 523 - 420

Calculating:

-211 - 267 = -478

-478 + 523 = 45

45 - 420 = -375

So $ x_7 = -375 $

Then $ x_8 = x_7 - x_6 + x_5 - x_4 $

-375 - (-211) + 267 - 523

Simplify:

-375 + 211 = -164

-164 + 267 = 103

103 - 523 = -420

So $ x_8 = -420 $

Next, $ x_9 = x_8 - x_7 + x_6 - x_5 $

-420 - (-375) + (-211) - 267

Calculating:

-420 + 375 = -45

-45 - 211 = -256

-256 - 267 = -523

So $ x_9 = -523 $

Now $ x_{10} = x_9 - x_8 + x_7 - x_6 $

-523 - (-420) + (-375) - (-211)

Simplify:

-523 + 420 = -103

-103 - 375 = -478

-478 + 211 = -267

So $ x_{10} = -267 $

Moving on to $ x_{11} = x_{10} - x_9 + x_8 - x_7 $

-267 - (-523) + (-420) - (-375)

Calculating:

-267 + 523 = 256

256 - 420 = -164

-164 + 375 = 211

So $ x_{11} = 211 $

Wait a second! $ x_{11} = 211 $, which is the same as $ x_1 $. Let me check if the following terms also repeat.

Compute $ x_{12} = x_{11} - x_{10} + x_9 - x_8 $

211 - (-267) + (-523) - (-420)

Simplify:

211 + 267 = 478

478 - 523 = -45

-45 + 420 = 375

So $ x_{12} = 375 $, which is $ x_2 $.

Then $ x_{13} = x_{12} - x_{11} + x_{10} - x_9 $

375 - 211 + (-267) - (-523)

Calculating:

375 - 211 = 164

164 - 267 = -103

-103 + 523 = 420

So $ x_{13} = 420 $, which is $ x_3 $.

Next, $ x_{14} = x_{13} - x_{12} + x_{11} - x_{10} $

420 - 375 + 211 - (-267)

Calculating:

420 - 375 = 45

45 + 211 = 256

256 + 267 = 523

So $ x_{14} = 523 $, which is $ x_4 $.

Then $ x_{15} = x_{14} - x_{13} + x_{12} - x_{11} $

523 - 420 + 375 - 211

Wait, that's exactly the same as $ x_5 $, which was 267. So $ x_{15} = 267 $, same as $ x_5 $.

So it seems like the sequence is periodic with period 10. Let me check:

Original terms:

x1=211, x2=375, x3=420, x4=523, x5=267, x6=-211, x7=-375, x8=-420, x9=-523, x10=-267, x11=211, x12=375,...

Yes! So starting from x1, the sequence repeats every 10 terms. Therefore, the period is 10.

So the sequence is periodic with period 10. Therefore, to find x_n, we can compute n modulo 10, and then find the corresponding term in the first 10 terms.

But let me confirm this periodicity. Let me check a few more terms to be sure.

x11=211 (same as x1)

x12=375 (same as x2)

x13=420 (same as x3)

x14=523 (same as x4)

x15=267 (same as x5)

x16= x15 - x14 + x13 - x12 = 267 - 523 + 420 - 375 = same as x6: -211

Yes, same as x6. So the pattern continues. Therefore, the period is indeed 10.

Therefore, the sequence repeats every 10 terms. Therefore, for any n, x_n = x_{n mod 10}, but we have to be careful with the modulus. Let me check:

For example, n=11: 11 mod 10 = 1, so x11 = x1. Similarly, n=12: 12 mod 10 = 2, so x12 = x2. However, note that when n is a multiple of 10, say n=10, 10 mod 10 = 0. But in our case, the original terms start at n=1, so perhaps we need to adjust the modulus accordingly.

Wait, let me think. If the period is 10, then x_{k} = x_{k + 10} for all k ≥ 1. Therefore, to find x_n, we can compute n modulo 10, but if the remainder is 0, then it corresponds to the 10th term. For example:

If n=10, then 10 mod 10 = 0, but we need to map that to x10. So perhaps we can define:

Let m = n mod 10. If m = 0, then the term is x10; otherwise, it's x_m.

Alternatively, since the period is 10, the sequence is:

Term number: 1 2 3 4 5 6 7 8 9 10 11 12 ...

Value:        211, 375, 420, 523, 267, -211, -375, -420, -523, -267, 211, 375,...

So for term number n, if we compute (n - 1) mod 10, then:

For n=1: (1-1)=0 mod10=0 → index 0? But our first term is index 1. Maybe another way.

Alternatively, since the period is 10, then x_{n} = x_{n - 10} for n ≥ 11. Therefore, to find x_n, we can subtract 10 until we get into the first 10 terms.

Therefore, for any n, x_n = x_{n mod 10}, but if n mod 10 is 0, then it's x_{10}.

Alternatively, if we think of the sequence as starting at n=1, then the position within the period can be calculated as:

Let’s define k = n - 1. Then k mod 10 gives 0 to 9, which corresponds to the original terms 1 to 10. Therefore, x_n = x_{(k mod 10) + 1} where k = n - 1.

So for example, n=1: k=0, 0 mod10=0, so term is x_{0+1}=x1. Similarly, n=10: k=9, 9 mod10=9, term is x_{9+1}=x10. n=11: k=10, 10 mod10=0, term is x_{0+1}=x1. Perfect.

Therefore, in general, x_n = x_{(n - 1) mod 10 + 1}

Therefore, to compute x_n, we can compute (n - 1) mod 10, add 1, and get the position in the first 10 terms.

Therefore, for example, x_{531}:

Compute (531 - 1) mod 10 = 530 mod 10 = 0. Then add 1: 0 + 1 = 1. Therefore, x_{531} = x1 = 211.

Similarly, x_{753}:

(753 - 1) mod 10 = 752 mod 10 = 2. Add 1: 3. Therefore, x_{753} = x3 = 420.

Wait, hold on. Wait, let me check:

Wait, if (n - 1) mod 10 gives 0 to 9, then adding 1 gives 1 to 10. Therefore, if (n - 1) mod 10 = 0, then x_n = x1; if it's 1, x2; ... if it's 9, x10.

Therefore, for n=531:

(531 - 1) = 530. 530 divided by 10 is 53, remainder 0. Therefore, (n - 1) mod 10 = 0. Therefore, x_{531} = x1 = 211.

Similarly, n=753:

(753 - 1) = 752. 752 divided by 10 is 75*10=750, remainder 2. Therefore, (n - 1) mod 10 = 2. Therefore, x_{753} = x_{2 + 1} = x3? Wait, no:

Wait, if (n - 1) mod 10 = 2, then adding 1 gives 3? Wait, no:

Wait, the formula is x_n = x_{(n - 1) mod 10 + 1}

So if (n - 1) mod 10 = 0, then x_n = x_{0 + 1} = x1

If (n - 1) mod 10 = 1, then x_n = x_{1 + 1} = x2

...

If (n - 1) mod 10 = 9, then x_n = x_{9 + 1} = x10

Therefore, for n=753:

(n - 1) mod 10 = 752 mod 10 = 2. Therefore, x_{753} = x_{2 + 1} = x3? Wait, no:

Wait, (n - 1) mod 10 = 2, so the position is 2 + 1? Wait, no. Wait, the formula is:

Let me think again. Let me take n=1:

(n - 1) = 0, mod10=0, so position is 0 + 1 = 1. Correct.

n=2: (2 - 1)=1 mod10=1, position=1 + 1=2. Correct.

Wait, so the formula is x_n = x_{(n - 1) mod 10 + 1}

Therefore, if (n - 1) mod 10 = k, then x_n = x_{k + 1}

Therefore, for n=753:

k = (753 - 1) mod 10 = 752 mod 10 = 2. Therefore, x_{753} = x_{2 + 1} = x3. Which is 420. Correct.

Similarly, for n=975:

(975 - 1) = 974. 974 mod 10 = 4. Therefore, x_{975} = x_{4 + 1} = x5. Which is 267.

Wait, so then x_{531} + x_{753} + x_{975} = 211 + 420 + 267.

Let me compute that:

211 + 420 = 631; 631 + 267 = 898.

But wait, let me check again if my reasoning is correct. Because if the period is 10, then every 10 terms it repeats. Therefore, the value of x_n depends on n mod 10. However, I need to make sure that my calculation for each term is correct.

Let me check for n=10:

(n - 1) = 9, mod10=9, so x_{10} = x_{9 + 1} = x10. Correct.

n=11: (11 - 1)=10 mod10=0, so x_{11}=x_{0 + 1}=x1. Correct.

Therefore, the formula is correct.

So for n=531:

531 divided by 10 is 53*10 + 1. Therefore, 531 = 53*10 + 1. Therefore, the remainder is 1. Therefore, since the period is 10, x_{531} = x1 = 211. Correct.

Similarly, n=753:

Divide 753 by 10: 75*10 = 750, 753 - 750 = 3. Therefore, remainder is 3. Therefore, x_{753} = x3 = 420. Wait, but according to previous calculation, (n - 1) mod10 = 2. Wait, there's a discrepancy here. Wait, which is correct?

Wait, if n=753, then n mod10 is 3. But according to the formula, (n - 1) mod10 = 752 mod10=2. Therefore, x_{753} = x_{2 + 1} = x3. Which is same as n mod10=3, so x_{n} = x_{n mod10} if n mod10 ≠0, else x_{10}. Wait, but here n mod10=3, so x_{753}=x3. Which is same as the formula. So both ways, if I think of n mod10, then:

If n mod10 = 0, then x_n = x10, else x_n = x_{n mod10}

But in our case, for n=753, n mod10=3, so x_{753}=x3. Which is correct. So both methods are equivalent.

Therefore, perhaps another way to think is:

If the period is 10, then x_n = x_{n mod10} if n mod10 ≠0, else x_{10}

But in our case, since the first term is x1, so for n=10, which is 10 mod10=0, we have x_{10}.

Therefore, for n=531:

531 divided by 10 is 53*10 + 1, so remainder 1. Therefore, x_{531}=x1=211.

For n=753:

753 divided by 10 is 75*10 + 3, remainder 3. Therefore, x_{753}=x3=420.

For n=975:

975 divided by 10 is 97*10 + 5, remainder 5. Therefore, x_{975}=x5=267.

Therefore, adding them up: 211 + 420 + 267.

Let me compute that again:

211 + 420 = 631

631 + 267:

631 + 200 = 831

831 + 67 = 898

So total is 898.

But wait, let me check if my calculation of x5 is correct. Earlier, I calculated x5 as 267. Let me check again:

x5 = x4 - x3 + x2 - x1 = 523 - 420 + 375 - 211.

Compute step by step:

523 - 420 = 103

103 + 375 = 478

478 - 211 = 267. Yes, correct.

So x5=267. Therefore, x_{975}=x5=267. So the sum is 211 + 420 + 267 = 898.

But wait, let me check if there's a mistake in the periodicity. Because when I calculated up to x10, then x11=x1, so the period is 10. But let me check x10 and x_{20} to be sure.

Earlier, x10=-267. Then x11=211, x12=375,..., x20 would be same as x10, which is -267. Let me check x20:

But maybe it's tedious, but since the recurrence is linear and we saw that after 10 terms it repeats, and the recurrence is of order 4, but with period 10, which is longer than the order. However, since we saw that the sequence repeats after 10 terms, and the calculation for x11 to x20 would follow the same pattern, so I think the periodicity is correct.

Therefore, the answer should be 898. However, let me check once again for n=531, 753, 975.

Alternatively, maybe I can check with another approach. Let me compute n mod 10 for each term:

For 531:

Divide 531 by 10: 53*10=530, 531-530=1. So remainder 1. Therefore, x_{531}=x1=211.

For 753:

753 divided by 10: 75*10=750, 753-750=3. Remainder 3. Therefore, x_{753}=x3=420.

For 975:

975 divided by 10: 97*10=970, 975-970=5. Remainder 5. Therefore, x_{975}=x5=267.

Therefore, sum is 211 + 420 + 267 = 898. So that seems correct.

But let me check if there's a possibility that the period is different? For example, maybe the period is 20 or something else? Wait, but we saw that x11=x1, x12=x2,..., so the period is 10. So unless there is a mistake in my calculation of the terms, which I checked up to x14, which repeats x4, so seems correct.

Alternatively, maybe I can check the recurrence relation for the periodicity. Suppose that the sequence is periodic with period 10, then for all n ≥ 1, x_{n+10} = x_n. Let me check for n=1:

x_{11} = x1 = 211. Correct.

Similarly, x_{12}=x2=375. Correct.

So if this holds for all n, then the period is indeed 10. Therefore, the answer is 898.

But let me check another term. For example, x_{10} = -267. Then x_{20} should be equal to x_{10} = -267. Let me compute x_{10} = -267, then x_{11}=211, x_{12}=375, x_{13}=420, x_{14}=523, x_{15}=267, x_{16}=-211, x_{17}=-375, x_{18}=-420, x_{19}=-523, x_{20}= -267. Yes, same as x_{10}. Therefore, the period is indeed 10. Therefore, the conclusion is correct.

Therefore, the answer is 211 + 420 + 267 = 898.

But wait, let me check once again the calculation of the sum:

211 + 420 = 631. Then 631 + 267:

Let me compute 631 + 200 = 831, then 831 + 67 = 898. Yes.

Alternatively, 211 + 267 = 478, then 478 + 420 = 898. Same result.

Therefore, I think the answer is 898.

But just to be thorough, let me check if there's a possibility that the problem is in the indexing. For example, sometimes sequences start at n=0, but here it's given as n=1, so I think the calculation is correct.

Alternatively, maybe I made a mistake in the initial terms? Let me check again:

x1=211, x2=375, x3=420, x4=523.

x5 = x4 - x3 + x2 - x1 = 523 - 420 + 375 - 211 = 523 - 420 is 103, 103 + 375 is 478, 478 - 211 is 267. Correct.

x6 = x5 - x4 + x3 - x2 = 267 - 523 + 420 - 375. 267 - 523 = -256, -256 + 420 = 164, 164 - 375 = -211. Correct.

x7 = x6 - x5 + x4 - x3 = -211 - 267 + 523 - 420. -211 -267 = -478, -478 + 523 = 45, 45 - 420 = -375. Correct.

x8 = x7 - x6 + x5 - x4 = -375 - (-211) + 267 - 523 = -375 + 211 = -164, -164 + 267 = 103, 103 - 523 = -420. Correct.

x9 = x8 - x7 + x6 - x5 = -420 - (-375) + (-211) - 267 = -420 + 375 = -45, -45 -211 = -256, -256 -267 = -523. Correct.

x10 = x9 - x8 + x7 - x6 = -523 - (-420) + (-375) - (-211) = -523 + 420 = -103, -103 - 375 = -478, -478 + 211 = -267. Correct.

x11 = x10 - x9 + x8 - x7 = -267 - (-523) + (-420) - (-375) = -267 + 523 = 256, 256 - 420 = -164, -164 + 375 = 211. Correct. So x11 = x1. All steps are correct.

Therefore, the periodicity is confirmed. Therefore, the answer is 898.

But just to be 100% sure, let me check another term. Let's compute x_{15} which should be equal to x5=267.

x11=211, x12=375, x13=420, x14=523, x15 = x14 - x13 + x12 - x11 = 523 - 420 + 375 - 211 = same as x5, which is 267. Correct.

Similarly, x_{16} = x15 - x14 + x13 - x12 = 267 - 523 + 420 - 375 = same as x6=-211. Correct.

Therefore, the periodicity is confirmed. Therefore, the answer is 898.

But wait, the problem says "find the value of $ x_{531} + x_{753} + x_{975} $." So according to my calculation, it's 211 + 420 + 267 = 898. However, let me check once again the values for each term:

For x_{531}:

Since 531 divided by 10 is 53*10 + 1, so remainder 1. Therefore, x_{531} = x1 = 211. Correct.

For x_{753}:

753 divided by 10 is 75*10 + 3, remainder 3. Therefore, x_{753} = x3 = 420. Correct.

For x_{975}:

975 divided by 10 is 97*10 + 5, remainder 5. Therefore, x_{975} = x5 = 267. Correct.

Therefore, adding them up: 211 + 420 + 267 = 898. So I think that's the answer.

But just to be thorough, let me check if there's an alternative way to compute this, maybe using the recurrence relation properties or matrix exponentiation or something else, but since the period is 10, it's straightforward.

Alternatively, maybe there's a mistake in assuming the period is 10? Let me check for n=10, x_{10}=-267, and then x_{10 + 10}=x_{20}=-267. Let me compute x_{15}=267, x_{16}=-211, x_{17}=-375, x_{18}=-420, x_{19}=-523, x_{20}= -267. Yes, same as x_{10}. Therefore, the period is indeed 10.

Therefore, I think my answer is correct. Therefore, the value is 898.

**Final Answer**
\boxed{898}

To solve the problem, we are given a recurrence relation:

$$
\begin{align*}
x_1 &= 211, \\
x_2 &= 375, \\
x_3 &= 420, \\
x_4 &= 523, \\
x_n &= x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} \quad \text{for } n \geq 5.
\end{align*}
$$

We are asked to compute:

$$
x_{531} + x_{753} + x_{975}
$$

---

### Step 1: Identify the Pattern

We begin by computing the first few terms of the sequence to look for a pattern or periodicity.

$$
\begin{align*}
x_1 &= 211 \\
x_2 &= 375 \\
x_3 &= 420 \\
x_4 &= 523 \\
x_5 &= x_4 - x_3 + x_2 - x_1 = 523 - 420 + 375 - 211 = 267 \\
x_6 &= x_5 - x_4 + x_3 - x_2 = 267 - 523 + 420 - 375 = -211 \\
x_7 &= x_6 - x_5 + x_4 - x_3 = -211 - 267 + 523 - 420 = -375 \\
x_8 &= x_7 - x_6 + x_5 - x_4 = -375 - (-211) + 267 - 523 = -420 \\
x_9 &= x_8 - x_7 + x_6 - x_5 = -420 - (-375) + (-211) - 267 = -523 \\
x_{10} &= x_9 - x_8 + x_7 - x_6 = -523 - (-420) + (-375) - (-211) = -267 \\
x_{11} &= x_{10} - x_9 + x_8 - x_7 = -267 - (-523) + (-420) - (-375) = 211 \\
x_{12} &= x_{11} - x_{10} + x_9 - x_8 = 211 - (-267) + (-523) - (-420) = 375 \\
x_{13} &= x_{12} - x_{11} + x_{10} - x_9 = 375 - 211 + (-267) - (-523) = 420 \\
x_{14} &= x_{13} - x_{12} + x_{11} - x_{10} = 420 - 375 + 211 - (-267) = 523 \\
x_{15} &= x_{14} - x_{13} + x_{12} - x_{11} = 523 - 420 + 375 - 211 = 267 \\
\end{align*}
$$

We observe that:

$$
x_{11} = x_1, \quad x_{12} = x_2, \quad x_{13} = x_3, \quad x_{14} = x_4, \quad x_{15} = x_5
$$

This confirms that the sequence is **periodic with period 10**.

---

### Step 2: Use the Periodicity to Compute the Required Terms

Since the sequence is periodic with period 10, we can compute any term $ x_n $ by finding the position of $ n $ within the first 10 terms.

To do this, we compute:

$$
n \mod 10
$$

- If the remainder is 0, then $ x_n = x_{10} $
- Otherwise, $ x_n = x_{\text{remainder}} $

Let’s compute each term:

- $ x_{531} $: $ 531 \mod 10 = 1 \Rightarrow x_{531} = x_1 = 211 $
- $ x_{753} $: $ 753 \mod 10 = 3 \Rightarrow x_{753} = x_3 = 420 $
- $ x_{975} $: $ 975 \mod 10 = 5 \Rightarrow x_{975} = x_5 = 267 $

---

### Step 3: Compute the Final Sum

$$
x_{531} + x_{753} + x_{975} = 211 + 420 + 267 = \boxed{898}
$$

---

### Final Answer

$$
\boxed{898}
$$

Solution: Shifting the recurrence over by one and adding, we have: \[\begin{aligned} x_n &= x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} \\ x_{n-1} &= x_{n-2} - x_{n-3} + x_{n-4} - x_{n-5} \\ \implies x_n + x_{n-1} &= x_{n-1} - x_{n-5} \end{aligned}\]so $x_n = -x_{n-5}$ for all $n.$ In particular, $x_n = -x_{n-5} = -(-x_{n-10}) = x_{n-10},$ so the sequence repeats with period $10.$ Thus, \[\begin{aligned} x_{531} + x_{753} + x_{975} &= x_1 + x_3 + x_5 \\ &= x_1 + x_3 + (x_4-x_3+x_2-x_1) \\ &= x_2 + x_4 \\ &= 375 + 523 = \boxed{898}. \end{aligned}\]