<Question>
Evaluate $\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil ) ^2$.
</Question>

<think>
<Plan_1>
Okay, let's see. I need to evaluate the expression ceiling of (3.6 squared) minus the ceiling of 3.6 squared. Hmm, let me break this down step by step. First, I should figure out what each part of the expression is.
The expression is ⎡(3.6)²⎤ - (⎡3.6⎤)². So, there are two main parts here: the ceiling of (3.6 squared) and the ceiling of 3.6 squared. Then subtract the second part from the first.

Let me start with the first part: (3.6)². What's 3.6 squared?

<prompt_1.1> Calculate 3.6 squared and apply the ceiling function to the result. </prompt_1.1>

Now the second part: (⎡3.6⎤)². First, find the ceiling of 3.6. The ceiling function, right?
Ceiling of 3.6 is the smallest integer greater than or equal to 3.6. Since 3.6 is between 3 and 4, the ceiling would be 4. Then, we square that result. So 4 squared is 16.
-----

Wait, that seems straightforward, but let me double-check each step to make sure I didn't make any mistakes.

First part: (3.6)^2.

<prompt_1.2> Recalculate 3.6 squared and verify the ceiling result. </prompt_1.2>

Second part: ceiling of 3.6 is 4, then 4 squared is 16.
-----

Is there any chance I mixed up the order of operations? Let me check the original expression again.

<prompt_1.3> Verify the interpretation of the original expression and consider alternative interpretations. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
3.6 times 3.6. Let me calculate that. 3 times 3 is 9, 3 times 0.6 is 1.8, 0.6 times 3 is 1.8, and 0.6 times 0.6 is 0.36. Adding those up: 9 + 1.8 + 1.8 + 0.36. Let's add the whole numbers first: 9 + 1 + 1 = 11. Then the decimals: 0.8 + 0.8 + 0.36 = 1.96. So total is 11 + 1.96 = 12.96. So 3.6 squared is 12.96.

Now, the ceiling function of 12.96. The ceiling function, denoted by ⎡x⎤, is the smallest integer greater than or equal to x. Since 12.96 is just a little less than 13, the ceiling would be 13. So the first part is 13.
</execution_1.1>

<execution_1.2>
3.6 multiplied by 3.6. Let me compute that. 3 times 3 is 9, 3 times 0.6 is 1.8, 0.6 times 3 is 1.8, and 0.6 times 0.6 is 0.36. Adding those up: 9 + 1.8 + 1.8 + 0.36. That's 9 + 3.6 + 0.36 = 12.96. So yes, (3.6)^2 is 12.96. Then the ceiling of 12.96. The ceiling function rounds up to the nearest integer. Since 12.96 is just 0.04 less than 13, the ceiling would be 13.
</execution_1.2>

<execution_1.3>
It's ceiling of (3.6 squared) minus (ceiling of 3.6) squared. So yes, order matters here. The first term is ceiling applied after squaring, and the second term is squaring applied after ceiling. So that's correct as per the problem statement.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Then the second part: ceiling of 3.6 is 4, as 3.6 is not an integer and the ceiling function rounds up to the nearest integer.
Then, squaring that result: 4 squared is 16. So the second part is 16.

Now subtract the second part from the first part: 13 - 16.
That equals -3.

So, putting it all together: ⎡(3.6)²⎤ - (⎡3.6⎤)² = 13 - 16 = -3.
-----

Is there a chance that the ceiling function was misapplied? Let me confirm.

<prompt_2.1> Confirm the application of the ceiling function in both parts of the expression. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
For the first part, the number is 12.96. Since ceiling rounds up to the nearest integer, 12.96 is between 12 and 13, so ceiling is 13. Correct. For the second part, 3.6's ceiling is 4, as 3.6 is between 3 and 4, so ceiling is 4. Squared gives 16. Correct.

Subtraction: 13 - 16. That's straightforward, results in -3.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but the problem is presented as ⎡(3.6)^2⎤ - (⎡3.6⎤)^2.
So, written out, that's ceiling(12.96) - (ceiling(3.6))^2 = 13 - 16 = -3.
-----

Therefore, the final answer should be -3. But let me just make sure that there isn't any notation confusion. Sometimes, different textbooks or resources might define the ceiling function differently, but generally, ceiling(x) is the smallest integer greater than or equal to x.

<prompt_3.1> Ensure there is no notation confusion regarding the ceiling function. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
So for positive numbers, that would round up to the next integer if it's not already an integer. Since 3.6 is not an integer, ceiling is 4. 12.96 is not an integer, so ceiling is 13. So, yes, all steps are correct.

Another way to think about it: maybe there's an alternative interpretation? For example, is the second term (ceiling of (3.6 squared)) or is it ceiling of (3.6) squared? But in the original problem, it's written as ( ⎡3.6⎤ )², so the ceiling is applied first, then squared. So, no ambiguity there. The first term is ceiling of the square, the second is square of the ceiling. So the calculation is correct.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the final answer is $\boxed{-3}$.
</Final_answer>
</think>