[THOUGHT] To find the probability that a regular cube's 12 edges can be colored with 12 different colors such that no two edges sharing a vertex have the same color, we need to consider the constraints and calculate both the total number of possible ways to color the edges (without constraints) and the number of valid ways to color them (with constraints).

First, observe that having 12 different colors for 12 edges means that each edge gets a unique color. The total number of ways to assign 12 different colors to 12 edges is simply 12 factorial (12!).

Next, we need to determine the number of valid colorings that meet the condition that no two edges sharing a vertex have the same color. Since we are using all 12 different colors, this is equivalent to finding the number of ways to arrange the colors such that no two adjacent edges share a color.

A key insight here is that a cube's edges can be partitioned into two sets of six edges such that no two edges within the same set share a vertex. This is because the cube can be decomposed into two disjoint sets of edges that form two "intertwined" hexagons (imagine the edges of two tensegrity structures). We can color each set independently with 6 distinct colors.

The number of ways to color one hexagon (set of 6 edges) is 6!. Since the two hexagons are independent, the total number of valid colorings is (6!)^2.

Thus, the probability is: [ \frac{\text{Number of valid colorings}}{\text{Total number of colorings}} = \frac{(6!)^2}{12!} ]

We can simplify this fraction by canceling common terms: [ \frac{6! \times 6!}{12!} = \frac{720 \times 720}{479001600} = \frac{518400}{479001600} = \frac{1}{9240} ]

Therefore, the fraction is (\frac{1}{9240}), and (p+q = 1 + 9240).

[/THOUGHT] [PYTHON] import math

Calculate the number of valid colorings

valid_colorings = (math.factorial(6)) ** 2

Calculate the total number of colorings

total_colorings = math.factorial(12)

Compute the probability as a fraction

probability_numerator = valid_colorings probability_denominator = total_colorings gcd = math.gcd(probability_numerator, probability_denominator) simplified_numerator = probability_numerator // gcd simplified_denominator = probability_denominator // gcd

Calculate p + q

p_plus_q = simplified_numerator + simplified_denominator print(p_plus_q)

[THOUGHT] The Python code confirms that the simplified probability of coloring the cube's edges with 12 different colors such that no two edges sharing a vertex have the same color is (\frac{1}{9240}). Therefore, (p = 9240) and (q = 1), and the sum (p + q = 925).

Thus, the final answer is: \boxed{925}