Explanation:
Okay, so I need to compute the cross product (2b - a) × (3c + a). The problem gives me the cross products of a×b, a×c, and b×c. Let me recall some cross product properties to see how I can use these given vectors.

First, cross product is distributive over vector addition, right? So, maybe I can expand the expression (2b - a) × (3c + a) using distributive property. Let me write that out:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a.

Simplifying each term:

First term: 2b × 3c = 6(b × c)
Second term: 2b × a = 2(b × a)
Third term: -a × 3c = -3(a × c)
Fourth term: -a × a = - (a × a)

Now, I remember that the cross product of any vector with itself is zero, so a × a = 0. So the fourth term is zero. That simplifies things.

So now, the expression becomes:

6(b × c) + 2(b × a) - 3(a × c)

But wait, cross product is anti-commutative, meaning that b × a = -a × b. So let me rewrite the second term:

2(b × a) = -2(a × b)

So substituting back:

6(b × c) - 2(a × b) - 3(a × c)

Now, the problem gives me the values of a×b, a×c, and b×c. Let me note them down:

Given:
a × b = [6, -7, 3]^T
a × c = [4, 7, 2]^T
b × c = [1, -7, 18]^T

So, substituting these into the expression:

First term: 6*(b × c) = 6*[1, -7, 18]^T = [6*1, 6*(-7), 6*18]^T = [6, -42, 108]^T

Second term: -2*(a × b) = -2*[6, -7, 3]^T = [-12, 14, -6]^T

Third term: -3*(a × c) = -3*[4, 7, 2]^T = [-12, -21, -6]^T

Now, I need to add these three vectors together:

First term: [6, -42, 108]
Second term: [-12, 14, -6]
Third term: [-12, -21, -6]

Let me add them component-wise.

For the first component (x-component):

6 + (-12) + (-12) = 6 - 12 -12 = 6 - 24 = -18

Second component (y-component):

-42 + 14 + (-21) = -42 + 14 -21 = (-42 -21) +14 = -63 +14 = -49

Third component (z-component):

108 + (-6) + (-6) = 108 -6 -6 = 108 -12 = 96

So putting it all together, the result is the vector [-18, -49, 96]^T.

Wait, let me check my calculations again to make sure I didn't make any arithmetic errors.

First component:

6 (from first term) -12 (second term) -12 (third term). 6 -12 is -6, then -6 -12 is -18. Correct.

Second component:

-42 (from first term) +14 (second term) -21 (third term). Let me compute step by step:

Start with -42 +14 = -28. Then -28 -21 = -49. Correct.

Third component:

108 (from first term) -6 (second term) -6 (third term). 108 -6 is 102, then 102 -6 is 96. Correct.

So the final vector is [-18, -49, 96]. Let me check if there's another way to approach this problem or if I missed any cross product properties.

Alternatively, maybe I can check if there's a different way to compute (2b - a) × (3c + a). Let me verify the expansion again.

Original expression: (2b - a) × (3c + a)

Expanding using distributive property:

2b × 3c + 2b × a - a × 3c - a × a

Which is 6(b × c) + 2(b × a) - 3(a × c) - 0

Then, since b × a = -a × b, so 2(b × a) = -2(a × b). So the expression becomes:

6(b × c) - 2(a × b) - 3(a × c). Which is exactly what I had before. So substitution is correct.

Therefore, substituting the given cross products:

6*(b × c) = 6*[1, -7, 18] = [6, -42, 108]

-2*(a × b) = -2*[6, -7, 3] = [-12, 14, -6]

-3*(a × c) = -3*[4, 7, 2] = [-12, -21, -6]

Adding them up:

x: 6 -12 -12 = -18

y: -42 +14 -21 = -49

z: 108 -6 -6 = 96

So the result is [-18, -49, 96]. Therefore, the answer should be this vector.

But let me check once more if I have the signs correct. Let me re-express the entire calculation:

Original expression:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a

Which is 6(b × c) + 2(b × a) - 3(a × c) - 0

Then, since b × a = -a × b, so 2(b × a) = -2(a × b). Therefore:

6(b × c) - 2(a × b) - 3(a × c)

Yes, that's correct. Then substituting:

6*(b × c) = 6*[1, -7, 18] = [6, -42, 108]

-2*(a × b) = -2*[6, -7, 3] = [-12, 14, -6]

-3*(a × c) = -3*[4, 7, 2] = [-12, -21, -6]

Adding them:

First component: 6 -12 -12 = -18

Second component: -42 +14 -21 = -49

Third component: 108 -6 -6 = 96

Yes, same result. So I think this is correct. Therefore, the answer is the vector (-18, -49, 96). Let me check if the problem expects the answer in a specific format, like boxed or something. The problem says to put the final answer within \boxed{}, so probably as a vector inside a box.

But let me check once again if there's a possibility of miscalculating the cross products. For example, maybe I mixed up the components?

Given:

a × b = [6, -7, 3]

a × c = [4, 7, 2]

b × c = [1, -7, 18]

So when I compute 6*(b × c), that's 6*1=6, 6*(-7)=-42, 6*18=108. Correct.

Then -2*(a × b): -2*6=-12, -2*(-7)=14, -2*3=-6. Correct.

Then -3*(a × c): -3*4=-12, -3*7=-21, -3*2=-6. Correct.

Adding them up:

x: 6 -12 -12 = -18

y: -42 +14 -21 = -49

z: 108 -6 -6 = 96

Yes, seems correct. Therefore, I think the answer is correct.

Alternatively, maybe I can check by computing the cross products in another way? For example, if I can find vectors a, b, c, then compute (2b -a) × (3c +a). But since we don't have the actual vectors, only their cross products, maybe that's not feasible. However, maybe we can check if the given cross products are consistent?

For example, if we have a×b, a×c, and b×c, can we find some relations?

Alternatively, maybe check if the cross products satisfy some properties. For example, the scalar triple product?

But since we don't have the dot products, maybe not. Alternatively, maybe check if the given cross products are consistent with each other?

Alternatively, maybe check if the determinant of the matrix formed by a, b, c is non-zero? But again, without knowing the actual vectors, it's hard to say.

Alternatively, maybe check if the cross products are orthogonal to the original vectors? For example, a×b is orthogonal to a and b. But since we don't have the actual vectors, maybe not helpful here.

Alternatively, maybe check if the answer makes sense. For example, the cross product (2b -a) × (3c +a) should be a vector that is orthogonal to both (2b -a) and (3c +a). But since we don't have the actual vectors, maybe not helpful.

Alternatively, maybe check if the components are integers, which they are, so that's a good sign.

Alternatively, maybe check if there's a miscalculation in the sign of the second term. Let me check again:

Original expansion:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a

Yes, that's correct. Then:

2b × 3c = 6(b × c)

2b × a = 2(b × a) = -2(a × b)

- a × 3c = -3(a × c)

- a × a = 0

So the expression is 6(b × c) - 2(a × b) - 3(a × c). Correct.

So substituting the given values:

Yes, that's correct. So I think my answer is correct.

Therefore, the final answer is \boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}.

**Final Answer**
\boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}

To compute the cross product
$$
(2\mathbf{b} - \mathbf{a}) \times (3\mathbf{c} + \mathbf{a}),
$$
we begin by applying the **distributive property** of the cross product over vector addition:

$$
(2\mathbf{b} - \mathbf{a}) \times (3\mathbf{c} + \mathbf{a}) = 2\mathbf{b} \times 3\mathbf{c} + 2\mathbf{b} \times \mathbf{a} - \mathbf{a} \times 3\mathbf{c} - \mathbf{a} \times \mathbf{a}.
$$

Now simplify each term:

- $ 2\mathbf{b} \times 3\mathbf{c} = 6(\mathbf{b} \times \mathbf{c}) $
- $ 2\mathbf{b} \times \mathbf{a} = 2(\mathbf{b} \times \mathbf{a}) = -2(\mathbf{a} \times \mathbf{b}) $ (since cross product is anti-commutative)
- $ -\mathbf{a} \times 3\mathbf{c} = -3(\mathbf{a} \times \mathbf{c}) $
- $ -\mathbf{a} \times \mathbf{a} = 0 $ (since the cross product of any vector with itself is zero)

So the expression becomes:

$$
6(\mathbf{b} \times \mathbf{c}) - 2(\mathbf{a} \times \mathbf{b}) - 3(\mathbf{a} \times \mathbf{c}).
$$

We are given the following cross products:

- $ \mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} $
- $ \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} $
- $ \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} $

Now compute each term:

1. $ 6(\mathbf{b} \times \mathbf{c}) = 6 \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} = \begin{pmatrix} 6 \\ -42 \\ 108 \end{pmatrix} $
2. $ -2(\mathbf{a} \times \mathbf{b}) = -2 \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} = \begin{pmatrix} -12 \\ 14 \\ -6 \end{pmatrix} $
3. $ -3(\mathbf{a} \times \mathbf{c}) = -3 \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} = \begin{pmatrix} -12 \\ -21 \\ -6 \end{pmatrix} $

Now add these three vectors component-wise:

- $ x $-component: $ 6 - 12 - 12 = -18 $
- $ y $-component: $ -42 + 14 - 21 = -49 $
- $ z $-component: $ 108 - 6 - 6 = 96 $

Thus, the final result is:

$$
\boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}
$$

[Assistant]
Explanation:
Okay, so I need to compute the cross product (2b - a) × (3c + a). The problem gives me the cross products of a×b, a×c, and b×c. Let me recall some cross product properties to see how I can use these given vectors.

First, cross product is distributive over vector addition, right? So, maybe I can expand the expression (2b - a) × (3c + a) using distributive property. Let me write that out:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a.

Simplifying each term:

First term: 2b × 3c = 6(b × c)
Second term: 2b × a = 2(b × a)
Third term: -a × 3c = -3(a × c)
Fourth term: -a × a = - (a × a)

Now, I remember that the cross product of any vector with itself is zero, so a × a = 0. So the fourth term is zero. That simplifies things.

So now, the expression becomes:

6(b × c) + 2(b × a) - 3(a × c)

But wait, cross product is anti-commutative, meaning that b × a = -a × b. So let me rewrite the second term:

2(b × a) = -2(a × b)

So substituting back:

6(b × c) - 2(a × b) - 3(a × c)

Now, the problem gives me the values of a×b, a×c, and b×c. Let me note them down:

Given:
a × b = [6, -7, 3]^T
a × c = [4, 7, 2]^T
b × c = [1, -7, 18]^T

So, substituting these into the expression:

First term: 6*(b × c) = 6*[1, -7, 18]^T = [6*1, 6*(-7), 6*18]^T = [6, -42, 108]^T

Second term: -2*(a × b) = -2*[6, -7, 3]^T = [-12, 14, -6]^T

Third term: -3*(a × c) = -3*[4, 7, 2]^T = [-12, -21, -6]^T

Now, I need to add these three vectors together:

First term: [6, -42, 108]
Second term: [-12, 14, -6]
Third term: [-12, -21, -6]

Let me add them component-wise.

For the first component (x-component):

6 + (-12) + (-12) = 6 - 12 -12 = 6 - 24 = -18

Second component (y-component):

-42 + 14 + (-21) = -42 + 14 -21 = (-42 -21) +14 = -63 +14 = -49

Third component (z-component):

108 + (-6) + (-6) = 108 -6 -6 = 108 -12 = 96

So putting it all together, the result is the vector [-18, -49, 96]^T.

Wait, let me check my calculations again to make sure I didn't make any arithmetic errors.

First component:

6 (from first term) -12 (second term) -12 (third term). 6 -12 is -6, then -6 -12 is -18. Correct.

Second component:

-42 (from first term) +14 (second term) -21 (third term). Let me compute step by step:

Start with -42 +14 = -28. Then -28 -21 = -49. Correct.

Third component:

108 (from first term) -6 (second term) -6 (third term). 108 -6 is 102, then 102 -6 is 96. Correct.

So the final vector is [-18, -49, 96]. Let me check if there's another way to approach this problem or if I missed any cross product properties.

Alternatively, maybe I can check if there's a different way to compute (2b - a) × (3c + a). Let me verify the expansion again.

Original expression: (2b - a) × (3c + a)

Expanding using distributive property:

2b × 3c + 2b × a - a × 3c - a × a

Which is 6(b × c) + 2(b × a) - 3(a × c) - 0

Then, since b × a = -a × b, so 2(b × a) = -2(a × b). So the expression becomes:

6(b × c) - 2(a × b) - 3(a × c). Which is exactly what I had before. So substitution is correct.

Therefore, substituting the given cross products:

6*(b × c) = 6*[1, -7, 18] = [6, -42, 108]

-2*(a × b) = -2*[6, -7, 3] = [-12, 14, -6]

-3*(a × c) = -3*[4, 7, 2] = [-12, -21, -6]

Adding them up:

x: 6 -12 -12 = -18

y: -42 +14 -21 = -49

z: 108 -6 -6 = 96

So the result is [-18, -49, 96]. Therefore, the answer should be this vector.

But let me check once more if I have the signs correct. Let me re-express the entire calculation:

Original expression:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a

Which is 6(b × c) + 2(b × a) - 3(a × c) - 0

Then, since b × a = -a × b, so 2(b × a) = -2(a × b). Therefore:

6(b × c) - 2(a × b) - 3(a × c)

Yes, that's correct. Then substituting:

6*(b × c) = 6*[1, -7, 18] = [6, -42, 108]

-2*(a × b) = -2*[6, -7, 3] = [-12, 14, -6]

-3*(a × c) = -3*[4, 7, 2] = [-12, -21, -6]

Adding them:

First component: 6 -12 -12 = -18

Second component: -42 +14 -21 = -49

Third component: 108 -6 -6 = 96

Yes, same result. So I think this is correct. Therefore, the answer is the vector (-18, -49, 96). Let me check if the problem expects the answer in a specific format, like boxed or something. The problem says to put the final answer within \boxed{}, so probably as a vector inside a box.

But let me check once again if there's a possibility of miscalculating the cross products. For example, maybe I mixed up the components?

Given:

a × b = [6, -7, 3]

a × c = [4, 7, 2]

b × c = [1, -7, 18]

So when I compute 6*(b × c), that's 6*1=6, 6*(-7)=-42, 6*18=108. Correct.

Then -2*(a × b): -2*6=-12, -2*(-7)=14, -2*3=-6. Correct.

Then -3*(a × c): -3*4=-12, -3*7=-21, -3*2=-6. Correct.

Adding them up:

x: 6 -12 -12 = -18

y: -42 +14 -21 = -49

z: 108 -6 -6 = 96

Yes, seems correct. Therefore, I think the answer is correct.

Alternatively, maybe I can check by computing the cross products in another way? For example, if I can find vectors a, b, c, then compute (2b -a) × (3c +a). But since we don't have the actual vectors, only their cross products, maybe that's not feasible. However, maybe we can check if the given cross products are consistent?

For example, if we have a×b, a×c, and b×c, can we find some relations?

Alternatively, maybe check if the cross products satisfy some properties. For example, the scalar triple product?

But since we don't have the dot products, maybe not. Alternatively, maybe check if the given cross products are consistent with each other?

Alternatively, maybe check if the determinant of the matrix formed by a, b, c is non-zero? But again, without knowing the actual vectors, it's hard to say.

Alternatively, maybe check if the cross products are orthogonal to the original vectors? For example, a×b is orthogonal to a and b. But since we don't have the actual vectors, maybe not helpful here.

Alternatively, maybe check if the answer makes sense. For example, the cross product (2b -a) × (3c +a) should be a vector that is orthogonal to both (2b -a) and (3c +a). But since we don't have the actual vectors, maybe not helpful.

Alternatively, maybe check if the components are integers, which they are, so that's a good sign.

Alternatively, maybe check if there's a miscalculation in the sign of the second term. Let me check again:

Original expansion:

(2b - a) × (3c + a) = 2b × 3c + 2b × a - a × 3c - a × a

Yes, that's correct. Then:

2b × 3c = 6(b × c)

2b × a = 2(b × a) = -2(a × b)

- a × 3c = -3(a × c)

- a × a = 0

So the expression is 6(b × c) - 2(a × b) - 3(a × c). Correct.

So substituting the given values:

Yes, that's correct. So I think my answer is correct.

Therefore, the final answer is \boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}.

**Final Answer**
\boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}

To compute the cross product
$$
(2\mathbf{b} - \mathbf{a}) \times (3\mathbf{c} + \mathbf{a}),
$$
we begin by applying the **distributive property** of the cross product over vector addition:

$$
(2\mathbf{b} - \mathbf{a}) \times (3\mathbf{c} + \mathbf{a}) = 2\mathbf{b} \times 3\mathbf{c} + 2\mathbf{b} \times \mathbf{a} - \mathbf{a} \times 3\mathbf{c} - \mathbf{a} \times \mathbf{a}.
$$

Now simplify each term:

- $ 2\mathbf{b} \times 3\mathbf{c} = 6(\mathbf{b} \times \mathbf{c}) $
- $ 2\mathbf{b} \times \mathbf{a} = 2(\mathbf{b} \times \mathbf{a}) = -2(\mathbf{a} \times \mathbf{b}) $ (since cross product is anti-commutative)
- $ -\mathbf{a} \times 3\mathbf{c} = -3(\mathbf{a} \times \mathbf{c}) $
- $ -\mathbf{a} \times \mathbf{a} = 0 $ (since the cross product of any vector with itself is zero)

So the expression becomes:

$$
6(\mathbf{b} \times \mathbf{c}) - 2(\mathbf{a} \times \mathbf{b}) - 3(\mathbf{a} \times \mathbf{c}).
$$

We are given the following cross products:

- $ \mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} $
- $ \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} $
- $ \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} $

Now compute each term:

1. $ 6(\mathbf{b} \times \mathbf{c}) = 6 \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} = \begin{pmatrix} 6 \\ -42 \\ 108 \end{pmatrix} $
2. $ -2(\mathbf{a} \times \mathbf{b}) = -2 \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix} = \begin{pmatrix} -12 \\ 14 \\ -6 \end{pmatrix} $
3. $ -3(\mathbf{a} \times \mathbf{c}) = -3 \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} = \begin{pmatrix} -12 \\ -21 \\ -6 \end{pmatrix} $

Now add these three vectors component-wise:

- $ x $-component: $ 6 - 12 - 12 = -18 $
- $ y $-component: $ -42 + 14 - 21 = -49 $
- $ z $-component: $ 108 - 6 - 6 = 96 $

Thus, the final result is:

$$
\boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}
$$

Solution:
Expanding, we get
\begin{align*}
(2 \mathbf{b} - \mathbf{a}) \times (3 \mathbf{c} + \mathbf{a}) &= 6 \mathbf{b} \times \mathbf{c} + 2 \mathbf{b} \times \mathbf{a} - 3 \mathbf{a} \times \mathbf{c} - \mathbf{a} \times \mathbf{a} \\
&= 6 \mathbf{b} \times \mathbf{c} - 2 \mathbf{a} \times \mathbf{b} - 3 \mathbf{a} \times \mathbf{c} - \mathbf{0} \\
&= 6 \begin{pmatrix} 1 \\ - 7 \\ 18 \end{pmatrix} - 2 \begin{pmatrix} 6 \\ - 7 \\ 3 \end{pmatrix} - 3 \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} \\
&= \boxed{\begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix}}.
\end{align*}


Explanation:
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 rectangle is centered at the origin, right? Because the x-coordinates go from -5 to 5 and the y-coordinates go from -4 to 4. So, it's a rectangle that's 10 units wide (from -5 to 5 on the x-axis) and 8 units tall (from -4 to 4 on the y-axis).

But the question is about integer coordinates strictly inside the rectangle. That means I need to count all the points (x, y) where x and y are integers, and the point is not on the boundary of the rectangle. So, the coordinates must satisfy -5 < x < 5 and -4 < y < 4. Wait, but since the rectangle's vertices are at those coordinates, the sides are at x = -5, x = 5, y = -4, and y = 4. So, strictly inside would mean x is between -5 and 5, not including the endpoints, and same with y.

But hold on, the problem says "strictly inside the rectangular region," so yes, the coordinates must be strictly inside. Therefore, for x, the integer values must be from -4 to 4 inclusive, because if x is -5 or 5, it's on the boundary. Similarly, for y, the integer values must be from -3 to 3 inclusive, because -4 and 4 are on the boundary.

Wait, let me check that. Let me think about the range for x. The rectangle spans from x = -5 to x = 5. So, the integer coordinates strictly inside would have x-values that are integers greater than -5 and less than 5. So, integers from -4 up to 4. Similarly for y, from -3 up to 3.

So, for x: how many integers are there between -4 and 4 inclusive? Let me calculate that. From -4 to 4, inclusive, the number of integers is 4 - (-4) + 1 = 4 + 4 + 1 = 9? Wait, no. Wait, the formula is (max - min) + 1. So, max is 4, min is -4. So, 4 - (-4) = 8, plus 1 is 9. So, 9 integer values for x. Similarly for y: from -3 to 3 inclusive. So, 3 - (-3) + 1 = 6 + 1 = 7. So, 7 integer values for y.

Therefore, the total number of integer coordinates strictly inside the rectangle would be 9 * 7 = 63. Is that right? Wait, let me check again.

But maybe I should think of it as for x: the original rectangle goes from -5 to 5. So, the interior would exclude x = -5 and x = 5. So, the integer x-values are -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Similarly, for y: original rectangle goes from -4 to 4, so interior excludes y = -4 and y = 4. So, integer y-values are -3, -2, -1, 0, 1, 2, 3. That's 7 values. So, 9 * 7 = 63.

But wait, let me make sure that I didn't make a mistake here. Let me think of a smaller rectangle to test. Suppose I have a rectangle from (0,0) to (2,2). Then the interior points would be (1,1). So, 1 point. Using the same method: x from 1 to 1 (since original x is 0 to 2, so interior is 0 < x < 2, so integer x is 1). Similarly y is 1. So, 1*1=1. Correct.

Another example: rectangle from (0,0) to (3,3). Then interior points would be x from 1,2 and y from 1,2. So, 2*2=4 points. Which are (1,1), (1,2), (2,1), (2,2). Correct. So, the formula seems to work.

So, applying that to our problem, 9*7=63. But let me check again.

Wait, but maybe I need to think about the coordinates. Let me count the number of integer x-coordinates strictly inside. The rectangle spans from x=-5 to x=5. So, the integer x-coordinates inside would be from -4 to 4 inclusive. How many numbers is that? Let's count them:

Starting from -4: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers. Similarly for y: from -3 to 3 inclusive. That's -3, -2, -1, 0, 1, 2, 3. That's 7 numbers. So, 9*7=63.

But wait, maybe I need to check if there's a different way to compute this. For example, the total number of integer points inside the rectangle can be calculated by (width - 1) * (height - 1). Wait, but here width is 10 units (from -5 to 5 is 10 units), but the number of integer points along the x-axis would be 10 + 1 = 11 points (including the endpoints). But if we exclude the endpoints, it's 11 - 2 = 9. Similarly, for the height: from -4 to 4 is 8 units, so number of integer points is 8 + 1 = 9. Excluding endpoints, it's 9 - 2 = 7. So, again, 9*7=63.

Alternatively, if I think of the rectangle as having width 10 and height 8, then the number of interior points would be (width - 2)*(height - 2). Wait, but that would be (10 - 2)*(8 - 2) = 8*6=48? Wait, that contradicts the previous answer. Wait, why is that?

Wait, maybe I confused the formula. Let me think again. If the rectangle is axis-aligned, then the number of integer points inside can be calculated by considering the number of integer points along each axis.

For example, if the rectangle spans from x = a to x = b, and y = c to y = d, then the number of integer points strictly inside would be (b - a - 1) * (d - c - 1). Wait, but in our case, the rectangle is from x=-5 to x=5, so b - a is 10, so b - a -1 is 9. Similarly, for y from -4 to 4, d - c is 8, so d - c -1 is 7. Therefore, 9*7=63. So that formula gives 63.

But earlier, when I thought of (width - 2)*(height - 2), that would be (10 - 2)*(8 - 2)=8*6=48. But that seems different. Wait, why is that?

Wait, maybe that formula is for something else. Let me think. If the rectangle is from x=0 to x=width, and y=0 to y=height, then the number of interior points would be (width - 1)*(height - 1). But in our case, the rectangle is from -5 to 5, which is 10 units wide. But if we think of it as starting at 0, then maybe not.

Alternatively, maybe the formula (width - 2)*(height - 2) is when you have a grid of points. Wait, perhaps I need to think of the number of integer points inside as (number of integer points along x-axis inside) multiplied by same for y-axis.

So, for the x-axis: from -5 to 5, the number of integer points is 11 (including endpoints). So, the number of interior points is 11 - 2 = 9. Similarly for y-axis: from -4 to 4, number of integer points is 9, so interior points is 9 - 2 = 7. Therefore, 9*7=63.

So, that seems correct. Therefore, the answer is 63? But let me check again with another approach.

Alternatively, think of the rectangle as having sides at x=-5, x=5, y=-4, y=4. The interior points are those where x is strictly between -5 and 5, and y strictly between -4 and 4. So, for x, integers from -4 to 4 inclusive. Let me count how many that is.

From -4 to 4 inclusive: that's 4 - (-4) + 1 = 9 numbers. Similarly for y: from -3 to 3 inclusive: 3 - (-3) + 1 = 7 numbers. So, 9*7=63.

Alternatively, maybe I can think of the total number of integer points in the rectangle including the boundary, and then subtract the boundary points.

Total number of integer points in the rectangle including boundary: for x from -5 to 5 inclusive, that's 11 points. For y from -4 to 4 inclusive, that's 9 points. So total points is 11*9=99.

Now, subtract the boundary points. But how many boundary points are there?

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

So, the total boundary points can be calculated as follows:

Top side: y=4, x from -5 to 5 inclusive. That's 11 points.

Bottom side: y=-4, x from -5 to 5 inclusive. That's 11 points.

Left side: x=-5, y from -4 to 4 inclusive. That's 9 points.

Right side: x=5, y from -4 to 4 inclusive. That's 9 points.

But then, the four corners are counted twice: each corner is on two sides. So, total boundary points would be 11 + 11 + 9 + 9 - 4 = 11 + 11 + 9 + 9 - 4 = 30 - 4 = 26? Wait, 11 + 11 is 22, 9 + 9 is 18, total 22 + 18 = 40, minus 4 corners counted twice, so 40 - 4 = 36.

Wait, so total boundary points are 36? Then total interior points would be total points (99) minus boundary points (36) = 63. Which matches the previous answer.

So, that seems to confirm that the answer is 63.

But let me check the boundary points again.

Top side: y=4, x from -5 to 5. That's 11 points. Similarly bottom side: 11 points.

Left side: x=-5, y from -4 to 4. That's 9 points. Similarly right side: 9 points.

But when we add them up, the four corners are counted twice. Each corner is on one top/bottom and one left/right. So, there are 4 corners. Therefore, total boundary points: 11 + 11 + 9 + 9 - 4 = 36.

So, total points in rectangle: 11*9=99.

Interior points: 99 - 36 = 63.

Yes, that seems correct.

Alternatively, maybe I can compute the number of boundary points another way.

The perimeter of the rectangle has four sides. But since it's a rectangle, the number of boundary points can be calculated as follows:

For the top and bottom sides: each has 11 points, but the corners are shared. However, when we calculated earlier, we considered top and bottom as separate, and left and right as separate.

Alternatively, the total number of boundary points is 2*(length + width) - 4. Wait, but that formula is for the number of lattice points on the perimeter? Wait, no, that formula is for the number of unit segments.

Wait, maybe not. Let me think.

Actually, the number of lattice points on the boundary can be calculated by considering each side.

For a rectangle with horizontal sides from x=a to x=b and vertical sides from y=c to y=d, the number of lattice points on the boundary is 2*( (b - a) + (d - c) ) - 4. Wait, let me check.

For example, take the rectangle from (0,0) to (2,2). The boundary points: top side (y=2): 3 points (x=0,1,2). Bottom side (y=0): 3 points. Left side (x=0): 3 points. Right side (x=2): 3 points. But corners are counted twice. So total boundary points: 3 + 3 + 3 + 3 - 4 = 8.

But according to the formula 2*( (b - a) + (d - c) ) - 4: here, b - a = 2 - 0 = 2, d - c = 2 - 0 = 2. So 2*(2 + 2) - 4 = 8 - 4 = 4? Which is not matching. So that formula is not correct.

Alternatively, maybe the formula is 2*(length + width) - 4? For the example above, length is 2, width is 2. 2*(2 + 2) - 4 = 8 - 4 = 4. But actual boundary points are 8. So that's not matching.

Alternatively, maybe the formula is 2*(length + width) - 4? Wait, no.

Alternatively, think of each side. For the top and bottom sides, each has (b - a + 1) points. Similarly for left and right sides, each has (d - c + 1) points. But then subtract 4 for the corners.

So, total boundary points: 2*(b - a + 1) + 2*(d - c + 1) - 4.

In our example, rectangle from (0,0) to (2,2): 2*(2 - 0 + 1) + 2*(2 - 0 + 1) - 4 = 2*3 + 2*3 - 4 = 6 + 6 - 4 = 8. Which matches.

So, applying this formula to our original problem.

Original rectangle: from x=-5 to x=5, so length along x-axis is 5 - (-5) = 10. Similarly, along y-axis from -4 to 4, length is 4 - (-4) = 8.

So, total boundary points would be 2*(10 + 1) + 2*(8 + 1) - 4. Wait, no. Wait, the formula is 2*(number of points on top and bottom) + 2*(number of points on left and right) - 4.

Wait, the number of points on top and bottom sides: each side has (length + 1) points? Wait, no. Wait, for the top and bottom sides, which are horizontal sides. The number of points on a horizontal side from x=a to x=b is (b - a + 1). Similarly for vertical sides.

In our case, the top side is from x=-5 to x=5, so number of points is 5 - (-5) + 1 = 11. Similarly for bottom side.

Left side is from y=-4 to y=4, which is 4 - (-4) + 1 = 9 points. Similarly for right side.

Therefore, total boundary points: 2*11 + 2*9 - 4 = 22 + 18 - 4 = 36. Which matches our previous calculation.

Therefore, total boundary points are 36.

Total points in rectangle (including boundary): 11*9=99.

Therefore, interior points: 99 - 36 = 63.

So, that seems consistent.

Alternatively, maybe I can think of the interior points as the number of integer points where x ranges from -4 to 4 inclusive and y ranges from -3 to 3 inclusive.

So, for x: from -4 to 4 inclusive. How many numbers is that? Let me count:

Starting at -4: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers.

For y: from -3 to 3 inclusive: -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

So, 9*7=63.

Therefore, regardless of the method, I get 63.

But just to be thorough, let me check with a different approach.

Imagine that for each x-coordinate from -4 to 4, how many y-coordinates are there? For each x, the y must be from -3 to 3 inclusive. So, 7 values.

So, for each of the 9 x-values, there are 7 y-values. So, 9*7=63.

Alternatively, think of the rectangle as a grid. The rectangle spans from x=-5 to x=5, which is 10 units wide, and y=-4 to y=4, which is 8 units tall.

But the number of interior points is (width - 2) * (height - 2). Wait, but here width is 10, so 10 - 2 = 8, height is 8, so 8 - 2 = 6. 8*6=48. Wait, that contradicts previous answers.

Wait, why is there a discrepancy here?

Wait, maybe this formula is for something else. Let me think. If the rectangle is from x=0 to x=width, and y=0 to y=height, then the number of interior points would be (width - 1)*(height - 1). But in our case, the rectangle is not starting at 0.

Alternatively, if we think of the number of interior points as (number of integer points along x-axis inside) * (number along y-axis inside). Which we already calculated as 9*7=63.

But why does the formula (width - 2)*(height - 2) give 8*6=48?

Wait, maybe that formula is when the rectangle is aligned with the grid such that the sides are at integer coordinates. For example, if the rectangle is from (0,0) to (n,m), then the number of interior points is (n - 1)*(m - 1). But in our case, the rectangle is from (-5, -4) to (5, 4). So, if we think of the width as 10 units (from -5 to 5), but the number of interior points along x-axis is 10 - 2 = 8? Wait, no. Wait, if you have a rectangle that spans from x = a to x = b, then the number of interior points along x-axis is (b - a - 1). Wait, if a and b are integers, then the number of integer points strictly inside is (b - a - 1).

Wait, for example, if a = -5 and b = 5, then b - a - 1 = 5 - (-5) - 1 = 10 - 1 = 9. Which matches our previous count.

Similarly, for y from -4 to 4, the number of interior points is 4 - (-4) - 1 = 8 - 1 = 7.

So, the formula is (b - a - 1) * (d - c - 1).

So, in this case, (5 - (-5) - 1) * (4 - (-4) - 1) = (10 - 1) * (8 - 1) = 9 * 7 = 63.

So, that formula gives the same answer.

Therefore, the formula (width - 1) * (height - 1) is not applicable here because width and height are measured in units, not in number of points.

So, the confusion was because of different interpretations of width and height.

Therefore, all methods lead to 63.

But just to be 100% sure, let me take a small example again.

Suppose we have a rectangle from (-1, -1) to (1, 1). So, vertices at (-1,-1), (-1,1), (1,-1), (1,1).

Strictly inside points would be x from -0 to 0 inclusive? Wait, no. Wait, strictly inside would be x between -1 and 1, not including the endpoints. So, integer x-values are 0. Similarly, y-values are 0. So, only (0,0). So, 1 point.

Using the formula: (b - a - 1) * (d - c - 1). Here, a=-1, b=1, c=-1, d=1.

So, (1 - (-1) - 1) * (1 - (-1) - 1) = (2 - 1) * (2 - 1) = 1*1=1. Correct.

Alternatively, total points in rectangle: (number of x points) * (number of y points). From x=-1 to 1 inclusive: 3 points. Similarly y: 3 points. Total 9. Boundary points: 4 sides.

Top side: y=1, x=-1,0,1: 3 points. Bottom side: same. Left side: x=-1, y=-1,0,1: 3 points. Right side: same. Total boundary points: 3 + 3 + 3 + 3 - 4 = 8. So, interior points: 9 - 8 = 1. Correct.

So, formula works here.

Another example: rectangle from (-2, -2) to (2, 2).

Strictly inside points: x from -1, 0, 1. y from -1, 0, 1. So, 3*3=9 points.

Using formula: (2 - (-2) -1) * (2 - (-2) -1) = (4 -1)*(4 -1)= 3*3=9. Correct.

Total points in rectangle: x from -2 to 2: 5 points. y same: 5 points. Total 25. Boundary points: 4 sides.

Top: y=2, x=-2,-1,0,1,2: 5 points. Bottom same. Left: x=-2, y=-2,-1,0,1,2: 5 points. Right same. Total boundary points: 5+5+5+5 - 4 = 20 - 4 = 16. Interior points: 25 - 16 = 9. Correct.

So, formula works.

Therefore, returning to original problem, answer is 63.

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

Imagine that for each row of y-values, how many x-values are there.

For y from -3 to 3 inclusive, that's 7 rows. For each row, the x-values are from -4 to 4 inclusive, which is 9 values. So, 7*9=63.

Yes.

Alternatively, think of the rectangle as a grid. The number of integer points inside is (number of columns - 2) * (number of rows - 2). Wait, but how does that relate?

Wait, if the rectangle has width W and height H, then the number of columns is W + 1, and number of rows is H + 1. But if we are talking about the number of columns between the boundaries, then maybe not.

Alternatively, maybe not.

But in any case, all methods lead to 63.

Therefore, I think the answer is 63.

But just to check once more, let me think of the coordinates.

For x: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers.

For y: -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

So, total points: 9*7=63.

Yes.

Therefore, I think the answer is 63.

**Final Answer**
The number of integer coordinates strictly inside the rectangular region is \boxed{63}.

To determine how many **integer coordinates** lie **strictly inside** the rectangle with vertices at $(5,4)$, $(5,-4)$, $(-5,4)$, and $(-5,-4)$, we begin by analyzing the bounds of the rectangle.

---

### Step 1: Understand the Rectangle's Dimensions

The rectangle is centered at the origin and spans:

- **Horizontally** from $x = -5$ to $x = 5$
- **Vertically** from $y = -4$ to $y = 4$

This makes the rectangle **10 units wide** (from $-5$ to $5$) and **8 units tall** (from $-4$ to $4$).

---

### Step 2: Determine the Range of Integer Coordinates Inside

We are asked to count **strictly inside** the rectangle, meaning the coordinates must **not lie on the boundary**. That is:

- $x$ must satisfy: $-5 < x < 5$
- $y$ must satisfy: $-4 < y < 4$

Since $x$ and $y$ must be **integers**, we find the integer values within these ranges:

- For $x$: The integers strictly between $-5$ and $5$ are:
  $-4, -3, -2, -1, 0, 1, 2, 3, 4$ → **9 values**
- For $y$: The integers strictly between $-4$ and $4$ are:
  $-3, -2, -1, 0, 1, 2, 3$ → **7 values**

---

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

Each valid $x$-coordinate can be paired with each valid $y$-coordinate. So the total number of integer coordinates strictly inside the rectangle is:

$$
9 \times 7 = 63
$$

---

### Step 4: Verification via Alternative Method

We can also verify this by computing the total number of integer points **within** the rectangle (including the boundary), and then subtracting the boundary points.

- Total integer points in the rectangle (including boundary):
  $x$ from $-5$ to $5$ → 11 values
  $y$ from $-4$ to $4$ → 9 values
  Total: $11 \times 9 = 99$

- Boundary points:
  - Top and bottom sides: $11$ points each → $2 \times 11 = 22$
  - Left and right sides: $9$ points each → $2 \times 9 = 18$
  - Subtract 4 corners (counted twice): $22 + 18 - 4 = 36$

- Interior points:
  $99 - 36 = 63$

---

### Final Answer

$$
\boxed{63}
$$

[Assistant]
Explanation:
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 rectangle is centered at the origin, right? Because the x-coordinates go from -5 to 5 and the y-coordinates go from -4 to 4. So, it's a rectangle that's 10 units wide (from -5 to 5 on the x-axis) and 8 units tall (from -4 to 4 on the y-axis).

But the question is about integer coordinates strictly inside the rectangle. That means I need to count all the points (x, y) where x and y are integers, and the point is not on the boundary of the rectangle. So, the coordinates must satisfy -5 < x < 5 and -4 < y < 4. Wait, but since the rectangle's vertices are at those coordinates, the sides are at x = -5, x = 5, y = -4, and y = 4. So, strictly inside would mean x is between -5 and 5, not including the endpoints, and same with y.

But hold on, the problem says "strictly inside the rectangular region," so yes, the coordinates must be strictly inside. Therefore, for x, the integer values must be from -4 to 4 inclusive, because if x is -5 or 5, it's on the boundary. Similarly, for y, the integer values must be from -3 to 3 inclusive, because -4 and 4 are on the boundary.

Wait, let me check that. Let me think about the range for x. The rectangle spans from x = -5 to x = 5. So, the integer coordinates strictly inside would have x-values that are integers greater than -5 and less than 5. So, integers from -4 up to 4. Similarly for y, from -3 up to 3.

So, for x: how many integers are there between -4 and 4 inclusive? Let me calculate that. From -4 to 4, inclusive, the number of integers is 4 - (-4) + 1 = 4 + 4 + 1 = 9? Wait, no. Wait, the formula is (max - min) + 1. So, max is 4, min is -4. So, 4 - (-4) = 8, plus 1 is 9. So, 9 integer values for x. Similarly for y: from -3 to 3 inclusive. So, 3 - (-3) + 1 = 6 + 1 = 7. So, 7 integer values for y.

Therefore, the total number of integer coordinates strictly inside the rectangle would be 9 * 7 = 63. Is that right? Wait, let me check again.

But maybe I should think of it as for x: the original rectangle goes from -5 to 5. So, the interior would exclude x = -5 and x = 5. So, the integer x-values are -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. Similarly, for y: original rectangle goes from -4 to 4, so interior excludes y = -4 and y = 4. So, integer y-values are -3, -2, -1, 0, 1, 2, 3. That's 7 values. So, 9 * 7 = 63.

But wait, let me make sure that I didn't make a mistake here. Let me think of a smaller rectangle to test. Suppose I have a rectangle from (0,0) to (2,2). Then the interior points would be (1,1). So, 1 point. Using the same method: x from 1 to 1 (since original x is 0 to 2, so interior is 0 < x < 2, so integer x is 1). Similarly y is 1. So, 1*1=1. Correct.

Another example: rectangle from (0,0) to (3,3). Then interior points would be x from 1,2 and y from 1,2. So, 2*2=4 points. Which are (1,1), (1,2), (2,1), (2,2). Correct. So, the formula seems to work.

So, applying that to our problem, 9*7=63. But let me check again.

Wait, but maybe I need to think about the coordinates. Let me count the number of integer x-coordinates strictly inside. The rectangle spans from x=-5 to x=5. So, the integer x-coordinates inside would be from -4 to 4 inclusive. How many numbers is that? Let's count them:

Starting from -4: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers. Similarly for y: from -3 to 3 inclusive. That's -3, -2, -1, 0, 1, 2, 3. That's 7 numbers. So, 9*7=63.

But wait, maybe I need to check if there's a different way to compute this. For example, the total number of integer points inside the rectangle can be calculated by (width - 1) * (height - 1). Wait, but here width is 10 units (from -5 to 5 is 10 units), but the number of integer points along the x-axis would be 10 + 1 = 11 points (including the endpoints). But if we exclude the endpoints, it's 11 - 2 = 9. Similarly, for the height: from -4 to 4 is 8 units, so number of integer points is 8 + 1 = 9. Excluding endpoints, it's 9 - 2 = 7. So, again, 9*7=63.

Alternatively, if I think of the rectangle as having width 10 and height 8, then the number of interior points would be (width - 2)*(height - 2). Wait, but that would be (10 - 2)*(8 - 2) = 8*6=48? Wait, that contradicts the previous answer. Wait, why is that?

Wait, maybe I confused the formula. Let me think again. If the rectangle is axis-aligned, then the number of integer points inside can be calculated by considering the number of integer points along each axis.

For example, if the rectangle spans from x = a to x = b, and y = c to y = d, then the number of integer points strictly inside would be (b - a - 1) * (d - c - 1). Wait, but in our case, the rectangle is from x=-5 to x=5, so b - a is 10, so b - a -1 is 9. Similarly, for y from -4 to 4, d - c is 8, so d - c -1 is 7. Therefore, 9*7=63. So that formula gives 63.

But earlier, when I thought of (width - 2)*(height - 2), that would be (10 - 2)*(8 - 2)=8*6=48. But that seems different. Wait, why is that?

Wait, maybe that formula is for something else. Let me think. If the rectangle is from x=0 to x=width, and y=0 to y=height, then the number of interior points would be (width - 1)*(height - 1). But in our case, the rectangle is from -5 to 5, which is 10 units wide. But if we think of it as starting at 0, then maybe not.

Alternatively, maybe the formula (width - 2)*(height - 2) is when you have a grid of points. Wait, perhaps I need to think of the number of integer points inside as (number of integer points along x-axis inside) multiplied by same for y-axis.

So, for the x-axis: from -5 to 5, the number of integer points is 11 (including endpoints). So, the number of interior points is 11 - 2 = 9. Similarly for y-axis: from -4 to 4, number of integer points is 9, so interior points is 9 - 2 = 7. Therefore, 9*7=63.

So, that seems correct. Therefore, the answer is 63? But let me check again with another approach.

Alternatively, think of the rectangle as having sides at x=-5, x=5, y=-4, y=4. The interior points are those where x is strictly between -5 and 5, and y strictly between -4 and 4. So, for x, integers from -4 to 4 inclusive. Let me count how many that is.

From -4 to 4 inclusive: that's 4 - (-4) + 1 = 9 numbers. Similarly for y: from -3 to 3 inclusive: 3 - (-3) + 1 = 7 numbers. So, 9*7=63.

Alternatively, maybe I can think of the total number of integer points in the rectangle including the boundary, and then subtract the boundary points.

Total number of integer points in the rectangle including boundary: for x from -5 to 5 inclusive, that's 11 points. For y from -4 to 4 inclusive, that's 9 points. So total points is 11*9=99.

Now, subtract the boundary points. But how many boundary points are there?

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

So, the total boundary points can be calculated as follows:

Top side: y=4, x from -5 to 5 inclusive. That's 11 points.

Bottom side: y=-4, x from -5 to 5 inclusive. That's 11 points.

Left side: x=-5, y from -4 to 4 inclusive. That's 9 points.

Right side: x=5, y from -4 to 4 inclusive. That's 9 points.

But then, the four corners are counted twice: each corner is on two sides. So, total boundary points would be 11 + 11 + 9 + 9 - 4 = 11 + 11 + 9 + 9 - 4 = 30 - 4 = 26? Wait, 11 + 11 is 22, 9 + 9 is 18, total 22 + 18 = 40, minus 4 corners counted twice, so 40 - 4 = 36.

Wait, so total boundary points are 36? Then total interior points would be total points (99) minus boundary points (36) = 63. Which matches the previous answer.

So, that seems to confirm that the answer is 63.

But let me check the boundary points again.

Top side: y=4, x from -5 to 5. That's 11 points. Similarly bottom side: 11 points.

Left side: x=-5, y from -4 to 4. That's 9 points. Similarly right side: 9 points.

But when we add them up, the four corners are counted twice. Each corner is on one top/bottom and one left/right. So, there are 4 corners. Therefore, total boundary points: 11 + 11 + 9 + 9 - 4 = 36.

So, total points in rectangle: 11*9=99.

Interior points: 99 - 36 = 63.

Yes, that seems correct.

Alternatively, maybe I can compute the number of boundary points another way.

The perimeter of the rectangle has four sides. But since it's a rectangle, the number of boundary points can be calculated as follows:

For the top and bottom sides: each has 11 points, but the corners are shared. However, when we calculated earlier, we considered top and bottom as separate, and left and right as separate.

Alternatively, the total number of boundary points is 2*(length + width) - 4. Wait, but that formula is for the number of lattice points on the perimeter? Wait, no, that formula is for the number of unit segments.

Wait, maybe not. Let me think.

Actually, the number of lattice points on the boundary can be calculated by considering each side.

For a rectangle with horizontal sides from x=a to x=b and vertical sides from y=c to y=d, the number of lattice points on the boundary is 2*( (b - a) + (d - c) ) - 4. Wait, let me check.

For example, take the rectangle from (0,0) to (2,2). The boundary points: top side (y=2): 3 points (x=0,1,2). Bottom side (y=0): 3 points. Left side (x=0): 3 points. Right side (x=2): 3 points. But corners are counted twice. So total boundary points: 3 + 3 + 3 + 3 - 4 = 8.

But according to the formula 2*( (b - a) + (d - c) ) - 4: here, b - a = 2 - 0 = 2, d - c = 2 - 0 = 2. So 2*(2 + 2) - 4 = 8 - 4 = 4? Which is not matching. So that formula is not correct.

Alternatively, maybe the formula is 2*(length + width) - 4? For the example above, length is 2, width is 2. 2*(2 + 2) - 4 = 8 - 4 = 4. But actual boundary points are 8. So that's not matching.

Alternatively, maybe the formula is 2*(length + width) - 4? Wait, no.

Alternatively, think of each side. For the top and bottom sides, each has (b - a + 1) points. Similarly for left and right sides, each has (d - c + 1) points. But then subtract 4 for the corners.

So, total boundary points: 2*(b - a + 1) + 2*(d - c + 1) - 4.

In our example, rectangle from (0,0) to (2,2): 2*(2 - 0 + 1) + 2*(2 - 0 + 1) - 4 = 2*3 + 2*3 - 4 = 6 + 6 - 4 = 8. Which matches.

So, applying this formula to our original problem.

Original rectangle: from x=-5 to x=5, so length along x-axis is 5 - (-5) = 10. Similarly, along y-axis from -4 to 4, length is 4 - (-4) = 8.

So, total boundary points would be 2*(10 + 1) + 2*(8 + 1) - 4. Wait, no. Wait, the formula is 2*(number of points on top and bottom) + 2*(number of points on left and right) - 4.

Wait, the number of points on top and bottom sides: each side has (length + 1) points? Wait, no. Wait, for the top and bottom sides, which are horizontal sides. The number of points on a horizontal side from x=a to x=b is (b - a + 1). Similarly for vertical sides.

In our case, the top side is from x=-5 to x=5, so number of points is 5 - (-5) + 1 = 11. Similarly for bottom side.

Left side is from y=-4 to y=4, which is 4 - (-4) + 1 = 9 points. Similarly for right side.

Therefore, total boundary points: 2*11 + 2*9 - 4 = 22 + 18 - 4 = 36. Which matches our previous calculation.

Therefore, total boundary points are 36.

Total points in rectangle (including boundary): 11*9=99.

Therefore, interior points: 99 - 36 = 63.

So, that seems consistent.

Alternatively, maybe I can think of the interior points as the number of integer points where x ranges from -4 to 4 inclusive and y ranges from -3 to 3 inclusive.

So, for x: from -4 to 4 inclusive. How many numbers is that? Let me count:

Starting at -4: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers.

For y: from -3 to 3 inclusive: -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

So, 9*7=63.

Therefore, regardless of the method, I get 63.

But just to be thorough, let me check with a different approach.

Imagine that for each x-coordinate from -4 to 4, how many y-coordinates are there? For each x, the y must be from -3 to 3 inclusive. So, 7 values.

So, for each of the 9 x-values, there are 7 y-values. So, 9*7=63.

Alternatively, think of the rectangle as a grid. The rectangle spans from x=-5 to x=5, which is 10 units wide, and y=-4 to y=4, which is 8 units tall.

But the number of interior points is (width - 2) * (height - 2). Wait, but here width is 10, so 10 - 2 = 8, height is 8, so 8 - 2 = 6. 8*6=48. Wait, that contradicts previous answers.

Wait, why is there a discrepancy here?

Wait, maybe this formula is for something else. Let me think. If the rectangle is from x=0 to x=width, and y=0 to y=height, then the number of interior points would be (width - 1)*(height - 1). But in our case, the rectangle is not starting at 0.

Alternatively, if we think of the number of interior points as (number of integer points along x-axis inside) * (number along y-axis inside). Which we already calculated as 9*7=63.

But why does the formula (width - 2)*(height - 2) give 8*6=48?

Wait, maybe that formula is when the rectangle is aligned with the grid such that the sides are at integer coordinates. For example, if the rectangle is from (0,0) to (n,m), then the number of interior points is (n - 1)*(m - 1). But in our case, the rectangle is from (-5, -4) to (5, 4). So, if we think of the width as 10 units (from -5 to 5), but the number of interior points along x-axis is 10 - 2 = 8? Wait, no. Wait, if you have a rectangle that spans from x = a to x = b, then the number of interior points along x-axis is (b - a - 1). Wait, if a and b are integers, then the number of integer points strictly inside is (b - a - 1).

Wait, for example, if a = -5 and b = 5, then b - a - 1 = 5 - (-5) - 1 = 10 - 1 = 9. Which matches our previous count.

Similarly, for y from -4 to 4, the number of interior points is 4 - (-4) - 1 = 8 - 1 = 7.

So, the formula is (b - a - 1) * (d - c - 1).

So, in this case, (5 - (-5) - 1) * (4 - (-4) - 1) = (10 - 1) * (8 - 1) = 9 * 7 = 63.

So, that formula gives the same answer.

Therefore, the formula (width - 1) * (height - 1) is not applicable here because width and height are measured in units, not in number of points.

So, the confusion was because of different interpretations of width and height.

Therefore, all methods lead to 63.

But just to be 100% sure, let me take a small example again.

Suppose we have a rectangle from (-1, -1) to (1, 1). So, vertices at (-1,-1), (-1,1), (1,-1), (1,1).

Strictly inside points would be x from -0 to 0 inclusive? Wait, no. Wait, strictly inside would be x between -1 and 1, not including the endpoints. So, integer x-values are 0. Similarly, y-values are 0. So, only (0,0). So, 1 point.

Using the formula: (b - a - 1) * (d - c - 1). Here, a=-1, b=1, c=-1, d=1.

So, (1 - (-1) - 1) * (1 - (-1) - 1) = (2 - 1) * (2 - 1) = 1*1=1. Correct.

Alternatively, total points in rectangle: (number of x points) * (number of y points). From x=-1 to 1 inclusive: 3 points. Similarly y: 3 points. Total 9. Boundary points: 4 sides.

Top side: y=1, x=-1,0,1: 3 points. Bottom side: same. Left side: x=-1, y=-1,0,1: 3 points. Right side: same. Total boundary points: 3 + 3 + 3 + 3 - 4 = 8. So, interior points: 9 - 8 = 1. Correct.

So, formula works here.

Another example: rectangle from (-2, -2) to (2, 2).

Strictly inside points: x from -1, 0, 1. y from -1, 0, 1. So, 3*3=9 points.

Using formula: (2 - (-2) -1) * (2 - (-2) -1) = (4 -1)*(4 -1)= 3*3=9. Correct.

Total points in rectangle: x from -2 to 2: 5 points. y same: 5 points. Total 25. Boundary points: 4 sides.

Top: y=2, x=-2,-1,0,1,2: 5 points. Bottom same. Left: x=-2, y=-2,-1,0,1,2: 5 points. Right same. Total boundary points: 5+5+5+5 - 4 = 20 - 4 = 16. Interior points: 25 - 16 = 9. Correct.

So, formula works.

Therefore, returning to original problem, answer is 63.

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

Imagine that for each row of y-values, how many x-values are there.

For y from -3 to 3 inclusive, that's 7 rows. For each row, the x-values are from -4 to 4 inclusive, which is 9 values. So, 7*9=63.

Yes.

Alternatively, think of the rectangle as a grid. The number of integer points inside is (number of columns - 2) * (number of rows - 2). Wait, but how does that relate?

Wait, if the rectangle has width W and height H, then the number of columns is W + 1, and number of rows is H + 1. But if we are talking about the number of columns between the boundaries, then maybe not.

Alternatively, maybe not.

But in any case, all methods lead to 63.

Therefore, I think the answer is 63.

But just to check once more, let me think of the coordinates.

For x: -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 numbers.

For y: -3, -2, -1, 0, 1, 2, 3. That's 7 numbers.

So, total points: 9*7=63.

Yes.

Therefore, I think the answer is 63.

**Final Answer**
The number of integer coordinates strictly inside the rectangular region is \boxed{63}.

To determine how many **integer coordinates** lie **strictly inside** the rectangle with vertices at $(5,4)$, $(5,-4)$, $(-5,4)$, and $(-5,-4)$, we begin by analyzing the bounds of the rectangle.

---

### Step 1: Understand the Rectangle's Dimensions

The rectangle is centered at the origin and spans:

- **Horizontally** from $x = -5$ to $x = 5$
- **Vertically** from $y = -4$ to $y = 4$

This makes the rectangle **10 units wide** (from $-5$ to $5$) and **8 units tall** (from $-4$ to $4$).

---

### Step 2: Determine the Range of Integer Coordinates Inside

We are asked to count **strictly inside** the rectangle, meaning the coordinates must **not lie on the boundary**. That is:

- $x$ must satisfy: $-5 < x < 5$
- $y$ must satisfy: $-4 < y < 4$

Since $x$ and $y$ must be **integers**, we find the integer values within these ranges:

- For $x$: The integers strictly between $-5$ and $5$ are:
  $-4, -3, -2, -1, 0, 1, 2, 3, 4$ → **9 values**
- For $y$: The integers strictly between $-4$ and $4$ are:
  $-3, -2, -1, 0, 1, 2, 3$ → **7 values**

---

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

Each valid $x$-coordinate can be paired with each valid $y$-coordinate. So the total number of integer coordinates strictly inside the rectangle is:

$$
9 \times 7 = 63
$$

---

### Step 4: Verification via Alternative Method

We can also verify this by computing the total number of integer points **within** the rectangle (including the boundary), and then subtracting the boundary points.

- Total integer points in the rectangle (including boundary):
  $x$ from $-5$ to $5$ → 11 values
  $y$ from $-4$ to $4$ → 9 values
  Total: $11 \times 9 = 99$

- Boundary points:
  - Top and bottom sides: $11$ points each → $2 \times 11 = 22$
  - Left and right sides: $9$ points each → $2 \times 9 = 18$
  - Subtract 4 corners (counted twice): $22 + 18 - 4 = 36$

- Interior points:
  $99 - 36 = 63$

---

### Final Answer

$$
\boxed{63}
$$

Solution: Points with integer coordinates are called lattice points. The length of the rectangle is $5 - (-5) = 10$ units. There will be 9 lattice positions between the two vertical sides of the rectangle. The height of the rectangle is $4 - (-4) = 8$ units. There will be 7 lattice positions between the top and bottom of the rectangle. That's a total of $9 \times 7 = \boxed{63}$ lattice points.


Explanation:
Okay, so I need to figure out the maximum number of points where at least two of the four distinct circles intersect. Hmm, let me start by recalling what I know about circles intersecting.

First, two circles can intersect in at most two points, right? If they're the same circle, they have infinitely many points, but since the problem says four distinct circles, I guess we don't have to worry about that. So each pair of circles can intersect at two points.

Now, if there are four circles, how many pairs of circles are there? Let me think. The number of ways to choose two circles out of four is given by the combination formula C(n, 2) where n is 4. So that would be 4! / (2! * (4-2)!) = (4*3)/2 = 6. So there are six pairs of circles.

If each pair can intersect at two points, then the maximum number of intersection points would be 6 pairs * 2 points per pair = 12 points. But wait, is there a possibility that some intersections overlap or something? Like, maybe if three circles all intersect at the same point, that would reduce the total number? But the question says "at least two of the circles intersect," so even if three circles intersect at the same point, that point would count as an intersection for each pair. However, the problem is asking for the maximum number of points, so we need to arrange the circles so that no two pairs share an intersection point.

So, if we can arrange the four circles such that every pair intersects at two unique points, not shared with any other pair, then the total number would be 12. But is that possible? Let me think.

Let me start with two circles. They can intersect at two points. Now, add a third circle. To maximize intersections, the third circle should intersect each of the first two circles at two distinct points. So, the third circle would add 2 intersections with the first circle and 2 with the second, so 4 new points. Similarly, adding a fourth circle, it should intersect each of the three existing circles at two distinct points. So that would be 2*3=6 new points.

So total intersections would be 2 (from first pair) + 4 (from third circle) + 6 (from fourth circle) = 12. Wait, but hold on, when you add the third circle, you have to make sure that the intersections with the first two circles don't coincide with any existing intersections. Similarly, when adding the fourth circle, its intersections with the first three circles should not overlap with any previous intersections.

Is this possible? Let me visualize. Let me think of the first two circles intersecting at two points. Now, the third circle can be arranged so that it crosses each of the first two circles at two different points. For example, if the first two circles intersect at points A and B, then the third circle can pass through two new points on the first circle, say C and D, and two new points on the second circle, E and F. Wait, but a circle can't pass through two points on the first circle and two points on the second circle without overlapping? Or maybe it can?

Wait, actually, if you have two circles, the third circle can intersect each of them at two points. So, for example, imagine the first two circles intersecting at two points. Then, the third circle can be arranged so that it intersects the first circle at two points that are not the original intersection points, and similarly for the second circle. So, as long as the third circle is not passing through any of the existing intersection points, then those intersections would be unique.

Similarly, when adding the fourth circle, it can be arranged to intersect each of the three existing circles at two new points, not overlapping with any previous intersections.

Therefore, if all these intersections are unique, then the total number of intersection points would be 2 + 4 + 6 = 12.

But wait, let me check again. The first two circles: 2 points. Then the third circle intersects each of the first two circles at two points, so that's 2 + 2 = 4 new points. Then the fourth circle intersects each of the three existing circles at two points, so 3*2=6 new points. So total is 2 + 4 + 6 = 12.

Alternatively, since there are C(4,2)=6 pairs, each contributing 2 points, so 6*2=12. So that seems to check out.

But is there a case where some intersections are overlapping? For example, if three circles all intersect at the same point, then that single point would be counted three times in the total, but in reality, it's just one point. However, since we are looking for the maximum number of points, we need to avoid such overlaps.

Therefore, if we can arrange the four circles so that no two pairs share an intersection point, then 12 is achievable.

But is this possible? Let me think of an example.

Let me try to imagine or sketch mentally. Let me take the first two circles, intersecting at two points. Let me call them circle A and circle B. Now, circle C needs to intersect circle A at two points different from A and B, and intersect circle B at two points different from A and B. So, if I draw circle C such that it crosses circle A at two new points and circle B at two new points.

Then, circle D needs to intersect circle A at two new points, circle B at two new points, and circle C at two new points. So, if I can arrange circle D so that it doesn't pass through any of the previous intersection points.

Is there a way to do this?

Well, in general, given four circles, if you arrange them so that each new circle intersects all previous ones at two unique points, then yes.

But maybe there's some geometric constraint? For example, if you have three circles, each pair intersecting at two points, then adding a fourth circle that intersects each of them at two points...

Wait, but in reality, when you add the fourth circle, you have to make sure that it doesn't pass through any existing intersection points. But since there are already multiple points, is there a way to position the fourth circle?

Alternatively, maybe there's a mathematical way to confirm that 12 is possible.

Alternatively, maybe there's a formula for the maximum number of intersections.

In general, for n circles, the maximum number of intersections is 2*C(n,2) = n(n-1). Because each pair contributes two points.

So for n=4, that would be 4*3=12.

Therefore, the answer should be 12.

But let me check for smaller numbers to see if this formula holds.

For n=2, two circles: maximum 2 points. The formula gives 2*1=2. Correct.

For n=3, three circles: each pair intersects at two points, so 3 pairs, 6 points. Is that possible? Let me think. If three circles, each pair intersecting at two distinct points, with no three circles intersecting at the same point.

Yes, for example, take three circles where each pair intersects at two unique points. For example, imagine three circles where each one is arranged so that they intersect the others at different points.

So yes, 6 points.

Therefore, for n=4, 12 points.

But maybe there's a catch here? Wait, but in reality, when you have more circles, there might be some constraints. For example, if you have four circles, can you really have all pairs intersecting at two distinct points?

Wait, let me think of coordinates. Maybe I can try to construct such circles.

Let me try to define four circles with coordinates.

First, take circle 1: x² + y² = 1.

Circle 2: (x - 1)² + y² = 1.

These two circles intersect at two points. Let me calculate them.

Subtracting the equations:

(x - 1)² + y² - (x² + y²) = 1 - 1 => 0

Expanding (x - 1)² - x² = 0 => x² - 2x + 1 - x² = -2x + 1 = 0 => x = 1/2.

Substituting back into circle 1: (1/2)² + y² = 1 => y² = 3/4 => y = ±√3/2.

So the two intersection points are (1/2, √3/2) and (1/2, -√3/2).

Now, let me define circle 3. Let me choose a circle that intersects circle 1 at two new points and circle 2 at two new points.

Let me try circle 3: x² + (y - 1)² = 1.

Let me check intersections with circle 1:

x² + y² = 1

and

x² + (y - 1)² = 1

Subtracting the first equation from the second:

(y - 1)² - y² = 0 => y² - 2y + 1 - y² = -2y + 1 = 0 => y = 1/2.

Substituting back into circle 1: x² + (1/2)^2 = 1 => x² = 3/4 => x = ±√3/2.

So the intersection points are (√3/2, 1/2) and (-√3/2, 1/2).

Now check intersections with circle 2:

Circle 2: (x - 1)^2 + y^2 = 1

Circle 3: x² + (y - 1)^2 = 1

Subtracting circle 2 from circle 3:

x² + (y - 1)^2 - [(x - 1)^2 + y^2] = 0

Expanding:

x² + y² - 2y + 1 - [x² - 2x + 1 + y²] = 0

Simplify:

x² + y² - 2y + 1 - x² + 2x - 1 - y² = -2y + 2x = 0 => 2x - 2y = 0 => x = y

So substituting x = y into circle 2:

(x - 1)^2 + x^2 = 1

Expanding: x² - 2x + 1 + x² = 1 => 2x² - 2x + 1 = 1 => 2x² - 2x = 0 => 2x(x - 1) = 0 => x = 0 or x = 1

Therefore, when x = 0, y = 0; when x = 1, y = 1.

So the intersection points are (0, 0) and (1, 1).

Now, check if these points are new.

Original intersections between circle 1 and 2 were (1/2, ±√3/2). The intersections between circle 1 and 3 are (√3/2, 1/2) and (-√3/2, 1/2). The intersections between circle 2 and 3 are (0, 0) and (1, 1). None of these points overlap with previous ones.

So circle 3 adds 4 new points.

Now, let's add circle 4. Let me try to define circle 4 such that it intersects each of the previous three circles at two new points.

Let me think. Let me try to define circle 4 with equation (x - a)^2 + (y - b)^2 = r^2.

We need to choose a, b, r such that it intersects circle 1, circle 2, and circle 3 each at two new points.

This might be complex, but let me try.

First, let me note the existing intersection points:

Circle 1 and 2: (1/2, √3/2), (1/2, -√3/2)

Circle 1 and 3: (√3/2, 1/2), (-√3/2, 1/2)

Circle 2 and 3: (0, 0), (1, 1)

So, circle 4 must not pass through any of these points.

Let me try to define circle 4 such that it intersects circle 1 at two new points, circle 2 at two new points, and circle 3 at two new points.

Let me try to choose circle 4 to pass through some other points.

Alternatively, maybe I can use symmetry. Let me think.

Alternatively, maybe take a circle that is larger or smaller.

Alternatively, let me try to define circle 4 as x² + y² = 2.

Wait, circle 1 is x² + y² = 1, circle 4 is x² + y² = 2. These are concentric circles, so they don't intersect. So that's bad.

Alternatively, let me try circle 4: (x - 2)^2 + y^2 = 1.

This circle is centered at (2, 0) with radius 1. Let me check intersections with circle 1: x² + y² = 1 and (x - 2)^2 + y^2 = 1.

Subtracting equations: (x - 2)^2 - x² = 0 => x² -4x +4 -x² = -4x +4 = 0 => x = 1.

Substituting back into circle 1: 1 + y² = 1 => y = 0. So intersection at (1, 0). But circle 1 and circle 2 intersect at (1/2, ±√3/2). So (1, 0) is a new point.

But wait, circle 4 and circle 1 intersect at (1, 0). But since circle 4 is (x - 2)^2 + y^2 = 1, which is a circle centered at (2, 0) with radius 1, so it touches circle 1 at (1, 0). Wait, but two circles can intersect at two points. Wait, but in this case, solving gives only one solution? Wait, no. Wait, when we subtracted the equations, we found x = 1, but substituting back gives y = 0. So that's only one point? Wait, but two circles should intersect at two points unless they are tangent.

Wait, let me check. The distance between centers of circle 1 (0,0) and circle 4 (2,0) is 2 units. The radii are 1 and 1. So the distance between centers is equal to the sum of radii? Wait, no. The sum of radii is 1 + 1 = 2, which is equal to the distance between centers. Therefore, the circles are externally tangent, so they intersect at exactly one point. Therefore, circle 4 and circle 1 intersect at one point. But we wanted two points. So this is not good.

So, maybe I need to adjust the circle. Let me try another circle.

Let me try circle 4: (x - 1)^2 + (y - 1)^2 = 1.

So center at (1,1), radius 1. Let me check intersections with circle 1: x² + y² = 1 and (x - 1)^2 + (y - 1)^2 = 1.

Expanding the second equation: x² - 2x + 1 + y² - 2y + 1 = 1 => x² + y² - 2x - 2y + 2 = 1 => x² + y² - 2x - 2y + 1 = 0.

But since x² + y² = 1, substitute into the above equation: 1 - 2x - 2y + 1 = 0 => -2x - 2y + 2 = 0 => x + y = 1.

So the intersections lie on the line x + y = 1.

Substituting into circle 1: x² + (1 - x)^2 = 1 => x² + 1 - 2x + x² = 1 => 2x² - 2x + 1 = 1 => 2x² - 2x = 0 => 2x(x - 1) = 0 => x = 0 or x = 1.

Therefore, when x = 0, y = 1; when x = 1, y = 0.

So the intersection points are (0, 1) and (1, 0).

Now, check if these points are new.

Original intersections:

Circle 1 and 2: (1/2, ±√3/2)

Circle 1 and 3: (√3/2, 1/2), (-√3/2, 1/2)

Circle 2 and 3: (0, 0), (1, 1)

So (0, 1) and (1, 0) are new points.

Now check intersections between circle 4 and circle 2.

Circle 2: (x - 1)^2 + y^2 = 1

Circle 4: (x - 1)^2 + (y - 1)^2 = 1

Subtracting circle 2 from circle 4:

[(x - 1)^2 + (y - 1)^2] - [(x - 1)^2 + y^2] = 0 => (y - 1)^2 - y^2 = 0 => y² - 2y + 1 - y² = -2y + 1 = 0 => y = 1/2.

Substituting back into circle 2: (x - 1)^2 + (1/2)^2 = 1 => (x - 1)^2 = 3/4 => x - 1 = ±√3/2 => x = 1 ± √3/2.

Therefore, the intersection points are (1 + √3/2, 1/2) and (1 - √3/2, 1/2).

Are these new? Let me check.

Original intersections between circle 2 and 3 were (0, 0) and (1, 1). The intersections between circle 1 and 2 were (1/2, ±√3/2). The intersections between circle 1 and 3 were (√3/2, 1/2) and (-√3/2, 1/2). The intersections between circle 2 and 4 are (1 ± √3/2, 1/2).

So, for example, (1 + √3/2, 1/2) is a new point, as is (1 - √3/2, 1/2).

Now check intersections between circle 4 and circle 3.

Circle 3: x² + (y - 1)^2 = 1

Circle 4: (x - 1)^2 + (y - 1)^2 = 1

Subtracting circle 3 from circle 4:

[(x - 1)^2 + (y - 1)^2] - [x² + (y - 1)^2] = 0 => (x - 1)^2 - x² = 0 => x² - 2x + 1 - x² = -2x + 1 = 0 => x = 1/2.

Substituting back into circle 3: (1/2)^2 + (y - 1)^2 = 1 => 1/4 + (y - 1)^2 = 1 => (y - 1)^2 = 3/4 => y - 1 = ±√3/2 => y = 1 ± √3/2.

Therefore, the intersection points are (1/2, 1 + √3/2) and (1/2, 1 - √3/2).

Check if these are new.

Original intersections:

Circle 1 and 2: (1/2, ±√3/2)

Circle 1 and 3: (√3/2, 1/2), (-√3/2, 1/2)

Circle 2 and 3: (0, 0), (1, 1)

So (1/2, 1 + √3/2) and (1/2, 1 - √3/2) are new points.

Therefore, circle 4 intersects circle 1 at (0, 1) and (1, 0), circle 2 at (1 ± √3/2, 1/2), and circle 3 at (1/2, 1 ± √3/2). All of these are new points not overlapping with previous intersections.

Therefore, circle 4 adds 2 + 2 + 2 = 6 new points.

So total number of intersection points is:

First pair (1-2): 2 points

Second pair (1-3): 2 points

Third pair (2-3): 2 points

Fourth pair (1-4): 2 points

Fifth pair (2-4): 2 points

Sixth pair (3-4): 2 points

Total: 6 pairs * 2 = 12 points.

And all these points are distinct.

Therefore, it is possible to have four circles with 12 intersection points.

Therefore, the maximum number is 12.

But let me check once more if there's any possibility of overlapping. For example, could any of the intersections from circle 4 with the previous circles coincide with each other?

For example, the intersection points between circle 4 and circle 1 are (0, 1) and (1, 0). The intersections between circle 4 and circle 2 are (1 ± √3/2, 1/2). The intersections between circle 4 and circle 3 are (1/2, 1 ± √3/2).

None of these points are the same. For example, (0, 1) is different from (1 ± √3/2, 1/2) and (1/2, 1 ± √3/2). Similarly, (1, 0) is different.

Therefore, all 12 points are unique.

Therefore, the maximum number is indeed 12.

But wait, let me think again. Is there a case where adding the fourth circle might cause some intersections to coincide? For example, if the fourth circle is arranged in a way that it intersects two previous circles at the same point? But in our construction, we made sure that all intersections are unique.

Alternatively, maybe there's a general principle here. For n circles, the maximum number of intersections is 2*C(n, 2) = n(n - 1).

But I remember that for lines, the maximum number of intersections is C(n, 2), since each pair intersects once. For circles, since each pair can intersect twice, it's double that.

Therefore, for four circles, 4*3=12.

Therefore, the answer should be 12.

But let me check online if I can recall... Wait, no, I need to think on my own.

Alternatively, think of each circle can intersect with another circle at two points. So for four circles, each circle can intersect with three others, so 2*3=6 points per circle. But since each intersection is counted twice (once for each circle), the total would be (4*6)/2=12.

Yes, that also gives 12.

Therefore, this seems consistent.

Therefore, I think the answer is 12.

**Final Answer**
The maximum number of points where at least two of the circles intersect is \boxed{12}.

To determine the **maximum number of points where at least two of the four distinct circles intersect**, we begin by analyzing the geometric properties of circle intersections.

---

### Step 1: Understanding Circle Intersections

Two distinct circles in a plane can intersect in **at most two points**. This is a well-known geometric fact. If the circles are identical, they would overlap completely, but since the problem specifies **distinct** circles, we can safely assume that each pair of circles can intersect at **up to two unique points**.

---

### Step 2: Counting the Number of Pairs

We are given **four distinct circles**. The number of **unique pairs** of circles that can be formed from four circles is given by the combination formula:

$$
\binom{4}{2} = \frac{4 \times 3}{2} = 6
$$

So, there are **6 unique pairs** of circles.

---

### Step 3: Maximum Number of Intersections

Each of these 6 pairs can intersect at **up to 2 points**, assuming no overlap or shared intersection points. Therefore, the **maximum number of intersection points** is:

$$
6 \text{ pairs} \times 2 \text{ points per pair} = 12 \text{ points}
$$

---

### Step 4: Ensuring All Intersections Are Unique

To achieve this maximum, we must ensure that **no two pairs of circles share the same intersection point**. This is possible by carefully arranging the circles so that each pair intersects at **two distinct points**, and these points are **not shared** with any other pair.

This can be achieved by choosing circles with different centers and radii, and ensuring that their intersection points are all distinct. For example, by using coordinate geometry or constructing circles with specific equations, we can verify that such an arrangement is possible.

---

### Step 5: Verifying with an Example

Let’s consider a concrete example:

- Circle 1: $ x^2 + y^2 = 1 $
- Circle 2: $ (x - 1)^2 + y^2 = 1 $
- Circle 3: $ x^2 + (y - 1)^2 = 1 $
- Circle 4: $ (x - 1)^2 + (y - 1)^2 = 1 $

By solving the equations pairwise, we can find the intersection points. Each pair intersects at **two unique points**, and none of these points overlap with those from other pairs. This confirms that **12 distinct intersection points** are achievable.

---

### Final Conclusion

Since each of the 6 pairs of circles can intersect at 2 unique points, and we can arrange the circles to ensure all these points are distinct, the **maximum number of points where at least two of the circles intersect** is:

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the maximum number of points where at least two of the four distinct circles intersect. Hmm, let me start by recalling what I know about circles intersecting.

First, two circles can intersect in at most two points, right? If they're the same circle, they have infinitely many points, but since the problem says four distinct circles, I guess we don't have to worry about that. So each pair of circles can intersect at two points.

Now, if there are four circles, how many pairs of circles are there? Let me think. The number of ways to choose two circles out of four is given by the combination formula C(n, 2) where n is 4. So that would be 4! / (2! * (4-2)!) = (4*3)/2 = 6. So there are six pairs of circles.

If each pair can intersect at two points, then the maximum number of intersection points would be 6 pairs * 2 points per pair = 12 points. But wait, is there a possibility that some intersections overlap or something? Like, maybe if three circles all intersect at the same point, that would reduce the total number? But the question says "at least two of the circles intersect," so even if three circles intersect at the same point, that point would count as an intersection for each pair. However, the problem is asking for the maximum number of points, so we need to arrange the circles so that no two pairs share an intersection point.

So, if we can arrange the four circles such that every pair intersects at two unique points, not shared with any other pair, then the total number would be 12. But is that possible? Let me think.

Let me start with two circles. They can intersect at two points. Now, add a third circle. To maximize intersections, the third circle should intersect each of the first two circles at two distinct points. So, the third circle would add 2 intersections with the first circle and 2 with the second, so 4 new points. Similarly, adding a fourth circle, it should intersect each of the three existing circles at two distinct points. So that would be 2*3=6 new points.

So total intersections would be 2 (from first pair) + 4 (from third circle) + 6 (from fourth circle) = 12. Wait, but hold on, when you add the third circle, you have to make sure that the intersections with the first two circles don't coincide with any existing intersections. Similarly, when adding the fourth circle, its intersections with the first three circles should not overlap with any previous intersections.

Is this possible? Let me visualize. Let me think of the first two circles intersecting at two points. Now, the third circle can be arranged so that it crosses each of the first two circles at two different points. For example, if the first two circles intersect at points A and B, then the third circle can pass through two new points on the first circle, say C and D, and two new points on the second circle, E and F. Wait, but a circle can't pass through two points on the first circle and two points on the second circle without overlapping? Or maybe it can?

Wait, actually, if you have two circles, the third circle can intersect each of them at two points. So, for example, imagine the first two circles intersecting at two points. Then, the third circle can be arranged so that it intersects the first circle at two points that are not the original intersection points, and similarly for the second circle. So, as long as the third circle is not passing through any of the existing intersection points, then those intersections would be unique.

Similarly, when adding the fourth circle, it can be arranged to intersect each of the three existing circles at two new points, not overlapping with any previous intersections.

Therefore, if all these intersections are unique, then the total number of intersection points would be 2 + 4 + 6 = 12.

But wait, let me check again. The first two circles: 2 points. Then the third circle intersects each of the first two circles at two points, so that's 2 + 2 = 4 new points. Then the fourth circle intersects each of the three existing circles at two points, so 3*2=6 new points. So total is 2 + 4 + 6 = 12.

Alternatively, since there are C(4,2)=6 pairs, each contributing 2 points, so 6*2=12. So that seems to check out.

But is there a case where some intersections are overlapping? For example, if three circles all intersect at the same point, then that single point would be counted three times in the total, but in reality, it's just one point. However, since we are looking for the maximum number of points, we need to avoid such overlaps.

Therefore, if we can arrange the four circles so that no two pairs share an intersection point, then 12 is achievable.

But is this possible? Let me think of an example.

Let me try to imagine or sketch mentally. Let me take the first two circles, intersecting at two points. Let me call them circle A and circle B. Now, circle C needs to intersect circle A at two points different from A and B, and intersect circle B at two points different from A and B. So, if I draw circle C such that it crosses circle A at two new points and circle B at two new points.

Then, circle D needs to intersect circle A at two new points, circle B at two new points, and circle C at two new points. So, if I can arrange circle D so that it doesn't pass through any of the previous intersection points.

Is there a way to do this?

Well, in general, given four circles, if you arrange them so that each new circle intersects all previous ones at two unique points, then yes.

But maybe there's some geometric constraint? For example, if you have three circles, each pair intersecting at two points, then adding a fourth circle that intersects each of them at two points...

Wait, but in reality, when you add the fourth circle, you have to make sure that it doesn't pass through any existing intersection points. But since there are already multiple points, is there a way to position the fourth circle?

Alternatively, maybe there's a mathematical way to confirm that 12 is possible.

Alternatively, maybe there's a formula for the maximum number of intersections.

In general, for n circles, the maximum number of intersections is 2*C(n,2) = n(n-1). Because each pair contributes two points.

So for n=4, that would be 4*3=12.

Therefore, the answer should be 12.

But let me check for smaller numbers to see if this formula holds.

For n=2, two circles: maximum 2 points. The formula gives 2*1=2. Correct.

For n=3, three circles: each pair intersects at two points, so 3 pairs, 6 points. Is that possible? Let me think. If three circles, each pair intersecting at two distinct points, with no three circles intersecting at the same point.

Yes, for example, take three circles where each pair intersects at two unique points. For example, imagine three circles where each one is arranged so that they intersect the others at different points.

So yes, 6 points.

Therefore, for n=4, 12 points.

But maybe there's a catch here? Wait, but in reality, when you have more circles, there might be some constraints. For example, if you have four circles, can you really have all pairs intersecting at two distinct points?

Wait, let me think of coordinates. Maybe I can try to construct such circles.

Let me try to define four circles with coordinates.

First, take circle 1: x² + y² = 1.

Circle 2: (x - 1)² + y² = 1.

These two circles intersect at two points. Let me calculate them.

Subtracting the equations:

(x - 1)² + y² - (x² + y²) = 1 - 1 => 0

Expanding (x - 1)² - x² = 0 => x² - 2x + 1 - x² = -2x + 1 = 0 => x = 1/2.

Substituting back into circle 1: (1/2)² + y² = 1 => y² = 3/4 => y = ±√3/2.

So the two intersection points are (1/2, √3/2) and (1/2, -√3/2).

Now, let me define circle 3. Let me choose a circle that intersects circle 1 at two new points and circle 2 at two new points.

Let me try circle 3: x² + (y - 1)² = 1.

Let me check intersections with circle 1:

x² + y² = 1

and

x² + (y - 1)² = 1

Subtracting the first equation from the second:

(y - 1)² - y² = 0 => y² - 2y + 1 - y² = -2y + 1 = 0 => y = 1/2.

Substituting back into circle 1: x² + (1/2)^2 = 1 => x² = 3/4 => x = ±√3/2.

So the intersection points are (√3/2, 1/2) and (-√3/2, 1/2).

Now check intersections with circle 2:

Circle 2: (x - 1)^2 + y^2 = 1

Circle 3: x² + (y - 1)^2 = 1

Subtracting circle 2 from circle 3:

x² + (y - 1)^2 - [(x - 1)^2 + y^2] = 0

Expanding:

x² + y² - 2y + 1 - [x² - 2x + 1 + y²] = 0

Simplify:

x² + y² - 2y + 1 - x² + 2x - 1 - y² = -2y + 2x = 0 => 2x - 2y = 0 => x = y

So substituting x = y into circle 2:

(x - 1)^2 + x^2 = 1

Expanding: x² - 2x + 1 + x² = 1 => 2x² - 2x + 1 = 1 => 2x² - 2x = 0 => 2x(x - 1) = 0 => x = 0 or x = 1

Therefore, when x = 0, y = 0; when x = 1, y = 1.

So the intersection points are (0, 0) and (1, 1).

Now, check if these points are new.

Original intersections between circle 1 and 2 were (1/2, ±√3/2). The intersections between circle 1 and 3 are (√3/2, 1/2) and (-√3/2, 1/2). The intersections between circle 2 and 3 are (0, 0) and (1, 1). None of these points overlap with previous ones.

So circle 3 adds 4 new points.

Now, let's add circle 4. Let me try to define circle 4 such that it intersects each of the previous three circles at two new points.

Let me think. Let me try to define circle 4 with equation (x - a)^2 + (y - b)^2 = r^2.

We need to choose a, b, r such that it intersects circle 1, circle 2, and circle 3 each at two new points.

This might be complex, but let me try.

First, let me note the existing intersection points:

Circle 1 and 2: (1/2, √3/2), (1/2, -√3/2)

Circle 1 and 3: (√3/2, 1/2), (-√3/2, 1/2)

Circle 2 and 3: (0, 0), (1, 1)

So, circle 4 must not pass through any of these points.

Let me try to define circle 4 such that it intersects circle 1 at two new points, circle 2 at two new points, and circle 3 at two new points.

Let me try to choose circle 4 to pass through some other points.

Alternatively, maybe I can use symmetry. Let me think.

Alternatively, maybe take a circle that is larger or smaller.

Alternatively, let me try to define circle 4 as x² + y² = 2.

Wait, circle 1 is x² + y² = 1, circle 4 is x² + y² = 2. These are concentric circles, so they don't intersect. So that's bad.

Alternatively, let me try circle 4: (x - 2)^2 + y^2 = 1.

This circle is centered at (2, 0) with radius 1. Let me check intersections with circle 1: x² + y² = 1 and (x - 2)^2 + y^2 = 1.

Subtracting equations: (x - 2)^2 - x² = 0 => x² -4x +4 -x² = -4x +4 = 0 => x = 1.

Substituting back into circle 1: 1 + y² = 1 => y = 0. So intersection at (1, 0). But circle 1 and circle 2 intersect at (1/2, ±√3/2). So (1, 0) is a new point.

But wait, circle 4 and circle 1 intersect at (1, 0). But since circle 4 is (x - 2)^2 + y^2 = 1, which is a circle centered at (2, 0) with radius 1, so it touches circle 1 at (1, 0). Wait, but two circles can intersect at two points. Wait, but in this case, solving gives only one solution? Wait, no. Wait, when we subtracted the equations, we found x = 1, but substituting back gives y = 0. So that's only one point? Wait, but two circles should intersect at two points unless they are tangent.

Wait, let me check. The distance between centers of circle 1 (0,0) and circle 4 (2,0) is 2 units. The radii are 1 and 1. So the distance between centers is equal to the sum of radii? Wait, no. The sum of radii is 1 + 1 = 2, which is equal to the distance between centers. Therefore, the circles are externally tangent, so they intersect at exactly one point. Therefore, circle 4 and circle 1 intersect at one point. But we wanted two points. So this is not good.

So, maybe I need to adjust the circle. Let me try another circle.

Let me try circle 4: (x - 1)^2 + (y - 1)^2 = 1.

So center at (1,1), radius 1. Let me check intersections with circle 1: x² + y² = 1 and (x - 1)^2 + (y - 1)^2 = 1.

Expanding the second equation: x² - 2x + 1 + y² - 2y + 1 = 1 => x² + y² - 2x - 2y + 2 = 1 => x² + y² - 2x - 2y + 1 = 0.

But since x² + y² = 1, substitute into the above equation: 1 - 2x - 2y + 1 = 0 => -2x - 2y + 2 = 0 => x + y = 1.

So the intersections lie on the line x + y = 1.

Substituting into circle 1: x² + (1 - x)^2 = 1 => x² + 1 - 2x + x² = 1 => 2x² - 2x + 1 = 1 => 2x² - 2x = 0 => 2x(x - 1) = 0 => x = 0 or x = 1.

Therefore, when x = 0, y = 1; when x = 1, y = 0.

So the intersection points are (0, 1) and (1, 0).

Now, check if these points are new.

Original intersections:

Circle 1 and 2: (1/2, ±√3/2)

Circle 1 and 3: (√3/2, 1/2), (-√3/2, 1/2)

Circle 2 and 3: (0, 0), (1, 1)

So (0, 1) and (1, 0) are new points.

Now check intersections between circle 4 and circle 2.

Circle 2: (x - 1)^2 + y^2 = 1

Circle 4: (x - 1)^2 + (y - 1)^2 = 1

Subtracting circle 2 from circle 4:

[(x - 1)^2 + (y - 1)^2] - [(x - 1)^2 + y^2] = 0 => (y - 1)^2 - y^2 = 0 => y² - 2y + 1 - y² = -2y + 1 = 0 => y = 1/2.

Substituting back into circle 2: (x - 1)^2 + (1/2)^2 = 1 => (x - 1)^2 = 3/4 => x - 1 = ±√3/2 => x = 1 ± √3/2.

Therefore, the intersection points are (1 + √3/2, 1/2) and (1 - √3/2, 1/2).

Are these new? Let me check.

Original intersections between circle 2 and 3 were (0, 0) and (1, 1). The intersections between circle 1 and 2 were (1/2, ±√3/2). The intersections between circle 1 and 3 were (√3/2, 1/2) and (-√3/2, 1/2). The intersections between circle 2 and 4 are (1 ± √3/2, 1/2).

So, for example, (1 + √3/2, 1/2) is a new point, as is (1 - √3/2, 1/2).

Now check intersections between circle 4 and circle 3.

Circle 3: x² + (y - 1)^2 = 1

Circle 4: (x - 1)^2 + (y - 1)^2 = 1

Subtracting circle 3 from circle 4:

[(x - 1)^2 + (y - 1)^2] - [x² + (y - 1)^2] = 0 => (x - 1)^2 - x² = 0 => x² - 2x + 1 - x² = -2x + 1 = 0 => x = 1/2.

Substituting back into circle 3: (1/2)^2 + (y - 1)^2 = 1 => 1/4 + (y - 1)^2 = 1 => (y - 1)^2 = 3/4 => y - 1 = ±√3/2 => y = 1 ± √3/2.

Therefore, the intersection points are (1/2, 1 + √3/2) and (1/2, 1 - √3/2).

Check if these are new.

Original intersections:

Circle 1 and 2: (1/2, ±√3/2)

Circle 1 and 3: (√3/2, 1/2), (-√3/2, 1/2)

Circle 2 and 3: (0, 0), (1, 1)

So (1/2, 1 + √3/2) and (1/2, 1 - √3/2) are new points.

Therefore, circle 4 intersects circle 1 at (0, 1) and (1, 0), circle 2 at (1 ± √3/2, 1/2), and circle 3 at (1/2, 1 ± √3/2). All of these are new points not overlapping with previous intersections.

Therefore, circle 4 adds 2 + 2 + 2 = 6 new points.

So total number of intersection points is:

First pair (1-2): 2 points

Second pair (1-3): 2 points

Third pair (2-3): 2 points

Fourth pair (1-4): 2 points

Fifth pair (2-4): 2 points

Sixth pair (3-4): 2 points

Total: 6 pairs * 2 = 12 points.

And all these points are distinct.

Therefore, it is possible to have four circles with 12 intersection points.

Therefore, the maximum number is 12.

But let me check once more if there's any possibility of overlapping. For example, could any of the intersections from circle 4 with the previous circles coincide with each other?

For example, the intersection points between circle 4 and circle 1 are (0, 1) and (1, 0). The intersections between circle 4 and circle 2 are (1 ± √3/2, 1/2). The intersections between circle 4 and circle 3 are (1/2, 1 ± √3/2).

None of these points are the same. For example, (0, 1) is different from (1 ± √3/2, 1/2) and (1/2, 1 ± √3/2). Similarly, (1, 0) is different.

Therefore, all 12 points are unique.

Therefore, the maximum number is indeed 12.

But wait, let me think again. Is there a case where adding the fourth circle might cause some intersections to coincide? For example, if the fourth circle is arranged in a way that it intersects two previous circles at the same point? But in our construction, we made sure that all intersections are unique.

Alternatively, maybe there's a general principle here. For n circles, the maximum number of intersections is 2*C(n, 2) = n(n - 1).

But I remember that for lines, the maximum number of intersections is C(n, 2), since each pair intersects once. For circles, since each pair can intersect twice, it's double that.

Therefore, for four circles, 4*3=12.

Therefore, the answer should be 12.

But let me check online if I can recall... Wait, no, I need to think on my own.

Alternatively, think of each circle can intersect with another circle at two points. So for four circles, each circle can intersect with three others, so 2*3=6 points per circle. But since each intersection is counted twice (once for each circle), the total would be (4*6)/2=12.

Yes, that also gives 12.

Therefore, this seems consistent.

Therefore, I think the answer is 12.

**Final Answer**
The maximum number of points where at least two of the circles intersect is \boxed{12}.

To determine the **maximum number of points where at least two of the four distinct circles intersect**, we begin by analyzing the geometric properties of circle intersections.

---

### Step 1: Understanding Circle Intersections

Two distinct circles in a plane can intersect in **at most two points**. This is a well-known geometric fact. If the circles are identical, they would overlap completely, but since the problem specifies **distinct** circles, we can safely assume that each pair of circles can intersect at **up to two unique points**.

---

### Step 2: Counting the Number of Pairs

We are given **four distinct circles**. The number of **unique pairs** of circles that can be formed from four circles is given by the combination formula:

$$
\binom{4}{2} = \frac{4 \times 3}{2} = 6
$$

So, there are **6 unique pairs** of circles.

---

### Step 3: Maximum Number of Intersections

Each of these 6 pairs can intersect at **up to 2 points**, assuming no overlap or shared intersection points. Therefore, the **maximum number of intersection points** is:

$$
6 \text{ pairs} \times 2 \text{ points per pair} = 12 \text{ points}
$$

---

### Step 4: Ensuring All Intersections Are Unique

To achieve this maximum, we must ensure that **no two pairs of circles share the same intersection point**. This is possible by carefully arranging the circles so that each pair intersects at **two distinct points**, and these points are **not shared** with any other pair.

This can be achieved by choosing circles with different centers and radii, and ensuring that their intersection points are all distinct. For example, by using coordinate geometry or constructing circles with specific equations, we can verify that such an arrangement is possible.

---

### Step 5: Verifying with an Example

Let’s consider a concrete example:

- Circle 1: $ x^2 + y^2 = 1 $
- Circle 2: $ (x - 1)^2 + y^2 = 1 $
- Circle 3: $ x^2 + (y - 1)^2 = 1 $
- Circle 4: $ (x - 1)^2 + (y - 1)^2 = 1 $

By solving the equations pairwise, we can find the intersection points. Each pair intersects at **two unique points**, and none of these points overlap with those from other pairs. This confirms that **12 distinct intersection points** are achievable.

---

### Final Conclusion

Since each of the 6 pairs of circles can intersect at 2 unique points, and we can arrange the circles to ensure all these points are distinct, the **maximum number of points where at least two of the circles intersect** is:

$$
\boxed{12}
$$

Solution:
Each pair of circles has at most two intersection points. There are $\binom{4}{2} = 6$ pairs of circles, so there are at most $6\times 2 = 12$ points of intersection. The following configuration shows that $\boxed{12}$ points of intersection are indeed possible:

[asy]
draw(Circle((0,0),2));
draw(Circle((1,1.3),2));
draw(Circle((-1,1.3),2));
draw(Circle((0,0.7),2));
[/asy]