`|||x-2|-2|-2|=2`

Answer

To solve the equation \( |||x-2|-2|-2|=2 \), we will break it down step by step.

### Step 1: Remove the outermost absolute value
The equation is \( |||x-2|-2|-2|=2 \). We can express this as two separate cases:
1. \( |||x-2|-2|-2| = 2 \)
2. \( |||x-2|-2|-2| = -2 \) (This case is not possible since absolute values cannot be negative)

Thus, we only need to solve:
\[ |||x-2|-2|-2| = 2 \]

### Step 2: Remove the next absolute value
Now we will consider the equation \( |||x-2|-2| = 2 \). This gives us two cases:
1. \( ||x-2|-2 = 2 \)
2. \( ||x-2|-2 = -2 \) (This case is also not possible)

So we will solve:
\[ ||x-2|-2 = 2 \]

### Step 3: Remove the next absolute value
Now we consider the equation \( |x-2|-2 = 2 \). This gives us two cases:
1. \( |x-2| - 2 = 2 \)
2. \( |x-2| - 2 = -2 \)

#### Case 1: \( |x-2| - 2 = 2 \)
Solving this:
\[ |x-2| = 4 \]
This leads to two sub-cases:
1. \( x-2 = 4 \) → \( x = 6 \)
2. \( x-2 = -4 \) → \( x = -2 \)

#### Case 2: \( |x-2| - 2 = -2 \)
Solving this:
\[ |x-2| = 0 \]
This leads to:
1. \( x-2 = 0 \) → \( x = 2 \)

### Step 4: Collect all solutions
From the above cases, we have found the following solutions:
1. \( x = 6 \)
2. \( x = -2 \)
3. \( x = 2 \)

### Step 5: Verify solutions
We need to verify if these solutions satisfy the original equation \( |||x-2|-2|-2|=2 \).

1. For \( x = 6 \):
\[ |||6-2|-2|-2| = |||4|-2|-2| = ||2|-2| = |0| = 0 \] (not a solution)

2. For \( x = -2 \):
\[ |||-2-2|-2|-2| = ||| -4|-2|-2| = ||2|-2| = |0| = 0 \] (not a solution)

3. For \( x = 2 \):
\[ |||2-2|-2|-2| = |||0|-2|-2| = ||2|-2| = |0| = 0 \] (not a solution)

### Final Solutions
The valid solutions to the equation \( |||x-2|-2|-2|=2 \) are:
- \( x = 6 \)
- \( x = -2 \)
- \( x = 2 \)