**Problem Statement:**

A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number?

**Solution:**

Let's assume the two-digit number to be "10x + y", where x and y represent the tens and units digits, respectively.

**Given Information:**

- The two-digit number is 4 more than 6 times the sum of its digits.
- If 18 is subtracted from the number, the digits are reversed.

**Step 1: Express the number in terms of its digits**

Based on the given information, we can express the two-digit number as follows:

"10x + y = 6(x + y) + 4"

**Step 2: Simplify the equation**

Let's simplify the equation obtained in Step 1:

10x + y = 6x + 6y + 4

**Step 3: Rearrange the equation**

To solve for the values of x and y, let's rearrange the equation:

10x - 6x = 6y - y + 4

4x = 5y + 4

**Step 4: Express x in terms of y**

Now, let's express x in terms of y by dividing both sides of the equation by 4:

x = (5y + 4)/4

**Step 5: Substitute the value of x in the equation**

Substitute the value of x from Step 4 into the original equation:

10((5y + 4)/4) + y = 6((5y + 4)/4) + 4

Simplify the equation:

(50y + 40)/4 + y = (30y + 24)/2 + 4

**Step 6: Solve the equation**

To solve the equation, let's multiply both sides by 4 to get rid of the denominators:

50y + 40 + 4y = 60y + 48 + 8

54y + 40 = 60y + 56

Subtract 54y from both sides:

40 = 6y + 56

Subtract 56 from both sides:

-16 = 6y

Divide both sides by 6:

y = -16/6

Simplify:

y = -8/3

But y represents the units digit, which should be a whole number. Since -8/3 is not a whole number, there is no solution to this equation.

Therefore, there is no two-digit number that satisfies the given conditions.