<Question>
Find the units digit of $18^6.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the units digit of 18^6. Hmm, let's see. Units digit problems usually have some pattern that repeats, right? So maybe if I can figure out the pattern in the units digits when powers of 18 are taken, I can determine the units digit of the 6th power.

<prompt_1.1> Identify the pattern in the units digits of powers of 18 and use it to determine the units digit of 18^6. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Wait, but 18 is a two-digit number, and the units digit of 18 is 8. So actually, the units digit of 18^6 will be the same as the units digit of 8^6, because when you multiply numbers, the units digit of the product depends only on the units digits of the factors. So 18^6 will have the same units digit as 8^6.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
That simplifies the problem to finding the units digit of 8^6. Now, let's recall the pattern for powers of 8.

<prompt_2.1> Recall the pattern for the units digits of powers of 8 and use it to determine the units digit of 8^6. </prompt_2.1>

Wait, let me verify that. Maybe I should compute 8^1 through 8^6 and check the units digits:

<prompt_2.2> Compute 8^1 through 8^6 and check the units digits to verify the pattern. </prompt_2.2>

But wait, let me make sure I didn't make a mistake in my calculations. Let's recheck each step.

<prompt_2.3> Recheck the calculations for 8^1 through 8^6 to ensure accuracy. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's compute the first few powers of 8 and see the units digits:

- 8^1 = 8 → units digit 8
- 8^2 = 64 → units digit 4
- 8^3 = 512 → units digit 2
- 8^4 = 4096 → units digit 6
- 8^5 = 32768 → units digit 8
- 8^6 = 262144 → units digit 4

So the units digits cycle every 4 exponents: 8, 4, 2, 6, and then repeats. So the cycle is 8, 4, 2, 6, 8, 4, 2, 6, etc.
</execution_2.1>

<execution_2.2>
8^1 = 8, units digit 8.

8^2 = 64, units digit 4.

8^3 = 512, units digit 2.

8^4 = 4096, units digit 6.

8^5 = 32768, units digit 8.

8^6 = 262144, units digit 4.
</execution_2.2>

<execution_2.3>
8^1 is 8, units digit 8. 8^2 is 64, units digit 4. 8^3 is 8*4 = 32, units digit 2. 8^4 is 8*2 = 16, units digit 6. 8^5 is 8*6 = 48, units digit 8. 8^6 is 8*8 = 64, units digit 4.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2, execution_2.3:
Alternatively, maybe there's a formula or a better way to calculate this without listing all the powers. For example, using modular arithmetic.

<prompt_3.1> Use modular arithmetic to find the units digit of 8^6 and confirm the result. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Since we're only interested in the units digit, which is equivalent to modulo 10. So 8^6 mod 10.

We can compute this using Euler's theorem or something. But since 8 and 10 are not coprime (they share a common divisor 2), Euler's theorem doesn't apply here. Hmm.

Alternatively, notice that 8 is congruent to -2 mod 10. So 8^6 ≡ (-2)^6 mod 10. (-2)^6 is 64, and 64 mod 10 is 4. So that gives the same result.

Alternatively, maybe break it down into exponents. For example, 8^6 = (8^2)^3. 8^2 is 64, which is 4 mod 10. Then 4^3 = 64, which is 4 mod 10.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Either way, the result is 4. So the units digit is 4.

But just to be thorough, let's compute 18^6 step by step, keeping only the units digit at each step, to confirm.

<prompt_4.1> Compute 18^6 step by step, keeping only the units digit at each step, to confirm the result. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
18^1 = 18 → units digit 8.

18^2 = 18*18 = 324 → units digit 4.

18^3 = 18^2 * 18 = 324*18. The units digit will be 4*8 = 32 → units digit 2.

18^4 = 18^3 * 18 = (units digit 2)*18. Units digit 2*8=16 → units digit 6.

18^5 = 18^4 * 18 = 6*8=48 → units digit 8.

18^6 = 18^5 * 18 = 8*8=64 → units digit 4.

So step by step, multiplying each time and taking the units digit, we get 8, 4, 2, 6, 8, 4 for the exponents 1 through 6.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the units digit of $18^6$ is $\boxed{4}$.
</Final_answer>
</think>