Let's denote the unknown number as \( x \).
According to the problem, one-fourth of the number is 2 more than one-sixth of the same number. We can express this relationship with the following equation:
\[ \frac{x}{4} = \frac{x}{6} + 2 \]
To solve for \( x \), we first eliminate the fractions by finding a common denominator. The least common multiple of 4 and 6 is 12, so we can multiply every term in the equation by 12:
\[ 12 \times \frac{x}{4} = 12 \times \left(\frac{x}{6} + 2\right) \]
This simplifies to:
\[ 3x = 2x + 24 \]
Next, we isolate \( x \) by subtracting \( 2x \) from both sides:
\[ 3x - 2x = 24 \]
This simplifies to:
\[ x = 24 \]
Thus, the number is \( \boxed{24} \).