<Question>
Consider the rectangle with vertices at $(5,4),$ $(5,-4),$ $(-5,4),$ $(-5,-4)$. How many integer coordinates will be strictly inside the rectangular region?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many integer coordinates 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 vertices are given, so the rectangle is centered at the origin, right? Because the x-coordinates go from -5 to 5, and the y-coordinates from -4 to 4.
Wait, but actually, the vertices are at (5,4), (5,-4), (-5,4), (-5,-4). So, the rectangle's sides are vertical lines at x=5 and x=-5, and horizontal lines at y=4 and y=-4. So, the rectangle includes all points (x,y) such that -5 ≤ x ≤ 5 and -4 ≤ y ≤ 4. But the question is asking for the integer coordinates strictly inside the rectangle. That means I need to exclude the boundary points, right?

So, first, let me recall that for a rectangle defined by x from a to b and y from c to d, the number of integer points inside (strictly) would be (b - a - 1) times (d - c - 1). Wait, is that correct? Let me think.

<prompt_1.1> Use the above plan to determine the number of integer coordinates strictly inside the rectangle. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
If we have x going from a to b, inclusive, the number of integers is b - a + 1. But if we want strictly inside, we need to exclude the endpoints. So, the x-values should be from a+1 to b-1. Similarly for y. So, the number of x-coordinates would be (b -1) - (a +1) +1 = b - a -1. Same for y. So, the total number of integer points inside would be (b - a -1) * (d - c -1).

Applying that here, the x goes from -5 to 5. So, a = -5, b =5. Then, the number of x-coordinates strictly inside would be 5 - (-5) -1 = 5 +5 -1 =9. Wait, let me check that again. If x is from -5 to 5 inclusive, the total number of integers is 5 - (-5) +1 =11. But strictly inside would exclude -5 and 5, so from -4 to 4 inclusive. So, how many integers is that? From -4 to 4, that's 4 - (-4) +1 =9. So yes, 9. So, the x part gives 9 possibilities.

Similarly for y. The y-values go from -4 to 4. The total number of integers is 4 - (-4) +1 =9. But strictly inside would exclude -4 and 4, so from -3 to 3 inclusive. The number of integers there is 3 - (-3) +1 =7. So, 7 possibilities.

Therefore, the total number of integer coordinates strictly inside the rectangle would be 9 *7 =63.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but let me verify this another way to make sure I'm not making a mistake.

Alternatively, maybe I can think of the rectangle as being in the coordinate system. The rectangle spans from x=-5 to x=5, so the length along the x-axis is 10 units (from -5 to 5 is 10 units). Similarly, along the y-axis, from y=-4 to y=4 is 8 units. But when counting integer points, the number of points is different.

<prompt_2.1> Verify the calculation of strictly interior integer coordinates using an alternative method. </prompt_2.1>

But let me visualize the grid. If I imagine the rectangle, the boundaries are at x=5 and x=-5, and y=4 and y=-4. So, any point with x-coordinate exactly 5 or -5 is on the boundary, same with y=4 or -4. So, strictly inside means x must be between -5 and 5, but not equal to either, so x can be -4, -3, ...., 3, 4. Similarly, y must be between -4 and 4, not equal to either, so y can be -3, -2, -1, 0, 1, 2, 3.

<prompt_2.2> Re-verify the strictly interior integer coordinates by listing possible x and y values. </prompt_2.2>
-----

Alternatively, maybe I can use the formula for the number of integer points in a rectangle. The general formula for the number of integer points (x,y) where a ≤ x ≤ b and c ≤ y ≤ d is (b - a +1)(d - c +1). So, applying that here, for the entire rectangle including boundaries, it's (5 - (-5) +1)(4 - (-4) +1) = (5 +5 +1)(4 +4 +1) = (11)(9) =99.

But this counts all the points from x=-5 to 5 and y=-4 to 4. But we want the strictly interior points, which would exclude the boundary. So, to get strictly interior, we need to subtract the boundary points. Alternatively, we can adjust the formula to exclude the boundaries.

<prompt_2.3> Use the inclusion method to verify the number of strictly interior integer coordinates. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
For example, along the x-axis from -5 to 5 inclusive, the integer x-coordinates are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. That's 11 points. Excluding the endpoints (-5 and 5), we have -4 to 4, which is 9 points. Similarly, along y-axis, from -4 to 4 inclusive is 9 points. Excluding endpoints gives -3 to 3, which is 7 points. So, 9*7=63.
</execution_2.1>

<execution_2.2>
So, how many x-coordinates is that? From -4 to 4 inclusive. Let's count: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 points. Similarly for y, from -3 to 3 inclusive. Let's count: -3, -2, -1, 0, 1, 2, 3. That's 7 points. So, 9 x-values and 7 y-values, so 9*7=63 points.
</execution_2.2>

<execution_2.3>
So, if we consider the interior as (-5 < x <5 and -4 < y <4), then x can be from -4 to 4 inclusive, which is 9 values, and y from -3 to 3 inclusive, which is 7 values, so 9*7=63 as before. So that seems consistent.

Alternatively, another way to think about it is that the original rectangle has 11 x-values (-5 to 5) and 9 y-values (-4 to 4), totaling 99 points. Then, the interior would be the rectangle from (-5 +1) to (5 -1) in x, and (-4 +1) to (4 -1) in y. So, x from -4 to 4, which is 9 points, and y from -3 to 3, which is 7 points. So, 9*7=63.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.2:
Alternatively, maybe I can list them out for x and y. Let's see.

<prompt_3.1> List possible x and y values to verify the strictly interior integer coordinates. </prompt_3.1>

Alternatively, maybe I can use the principle of inclusion here. The total number of integer points in the rectangle including boundaries is (5 - (-5) +1)(4 - (-4) +1) =11*9=99 as above. Then, subtract the points on the boundary. But that might be more complicated, but let's try.

<prompt_3.2> Verify the strictly interior integer coordinates using the inclusion method. </prompt_3.2>
-----

Based on execution_2.3:
Therefore, the number of integer coordinates strictly inside the rectangle is 63.

But just to make sure, let's take a small example.

<prompt_3.3> Test the method with smaller examples to ensure correctness. </prompt_3.3>
</Plan_3>

<Execution_3>
<execution_3.1>
For x strictly inside: the rectangle's left boundary is x=-5 and right boundary is x=5. So, strictly inside means x must be greater than -5 and less than 5. Since x is integer, that means x can be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Similarly, for y strictly inside: the top and bottom boundaries are y=4 and y=-4. So, y must be between -4 and 4, not including -4 and 4. So, y can be -3, -2, -1, 0, 1, 2, 3. That's 7 values. So, combining these, each x can pair with each y, so 9*7=63 points.
</execution_3.1>

<execution_3.2>
The boundary consists of four sides: left, right, top, bottom. But we have to be careful not to double subtract the corners.

Left side: x=-5, y from -4 to 4. That's 9 points. Right side: x=5, y from -4 to 4. Another 9 points. Top side: y=4, x from -5 to 5. That's 11 points. Bottom side: y=-4, x from -5 to5. Another 11 points. But the four corners ( (-5,-4), (-5,4), (5,-4), (5,4) ) are each counted twice, once in the vertical sides and once in the horizontal sides. So, total boundary points would be 9 +9 +11 +11 -4*2? Wait, no. Let's calculate:

Left and right sides: each have 9 points, so 2*9=18.

Top and bottom sides: each have 11 points, so 2*11=22.

But the four corners are included in both the vertical and horizontal sides. So, total boundary points would be 18 +22 - 4 (since the 4 corners are double-counted) =36.

Wait, let me check:

Left side: 9 points (y from -4 to 4)

Right side: 9 points

Top side: 11 points (x from -5 to5)

Bottom side:11 points

Total before subtracting overlaps: 9+9+11+11=40.

But the four corners are each counted twice, so subtract 4: 40 -4=36. Yes, that's correct. So, total boundary points are 36.

Therefore, the number of strictly interior points would be total points 99 - boundary points 36 =63. Which matches the previous result. So, that's a good check.

Therefore, the answer is 63.
</execution_3.2>

<execution_3.3>
Suppose we have a rectangle from x=1 to x=3, y=2 to y=4. Then, strictly inside would be x=2 and y=3. So, 1*1=1 point. Using the formula, (3-1 -1)*(4-2 -1)=(1)*(1)=1. Correct. If we use the total points including boundary: (3-1 +1)*(4-2 +1)=3*3=9. Then subtract the boundary points. The boundary would be all points where x=1,3 or y=2,4. Let's compute that. Points with x=1: y=2,3,4: 3 points. Points with x=3: y=2,3,4: 3 points. Points with y=2: x=1,2,3: 3 points. Points with y=4: x=1,2,3: 3 points. But subtract the corners twice. There are 4 corners: (1,2), (1,4), (3,2), (3,4). Each counted twice. So total boundary points: 3+3+3+3 -4=8. So, interior points would be 9 -8=1. Which matches. So, in this case, the formula works. So, similarly, in the original problem, 99 total points, subtract boundary points. But maybe that's more work. But since the direct counting gives 63, and the alternative method via subtracting boundaries also works in smaller cases, it's likely correct.
</execution_3.3>
</Execution_3>

<Plan_4>
Based on execution_3.2:
Another way to think about it is that when you have a rectangle from a to b inclusive, the number of interior points is (b - a -1). So, for the x-direction, from -5 to5, the number of interior x's is 5 - (-5) -1=11-1=10?

<prompt_4.1> Verify the strictly interior integer coordinates using the inclusion method. </prompt_4.1>
-----

Based on execution_3.3:
Alternatively, perhaps even list out the possible x and y. For x from -4 to4 inclusive, that's 9 numbers, and y from -3 to3 inclusive, 7 numbers. Each combination gives a unique point. 9*7=63.
So, that's straightforward.

Therefore, I think the answer is 63.
</Plan_4>

<Execution_4>
<execution_4.1>
Wait, that conflicts with the previous result where I thought it was 9. Wait, no. Wait, 5 - (-5) is 10, then subtract 1 gives 9. So, 10 -1=9. So, the formula is (b - a) -1. Wait, but in the general case, if you have integers from a to b inclusive, the number of integers is b - a +1. So, if we want to exclude the endpoints, then it's from a+1 to b-1, so the count is (b -1) - (a +1) +1 = b - a -1. So, yes, in this case, a=-5, b=5, so 5 - (-5) -1=10 -1=9. So, that's correct. Similarly, for y, from -4 to4 inclusive, excluding endpoints gives -3 to3, which is 3 - (-3) +1=7. So, 7 points. So, 9*7=63.
</execution_4.1>
</Execution_4>

<Final_answer>
Thus, the number of integer coordinates strictly inside the rectangular region is $\boxed{63}$.
</Final_answer>
</think>