Randomly draw three from a standard set of 28 Double Six Dominoes.

What is the probability they can be arranged in a triangle with each end touching another end with an equal number of spots?

For example: [1|3][3|0][0|1] can form such a triangle.

(Note: this triangle is not a legal configuration in an actual game of dominoes.)

Randomly draw four dominoes instead.

What is the probability of being able to form a square in the same fashion?

How about 5? 6? 27? 26?

There doesn't seem to be a general formula. Is there?

25 randomly drawn can form a 25-gon only if the undrawn three are all doubles, or if they form a triangle.

Probability that they are all doubles = 7/28 * 6/27 * 5/26 = 5/468

Probability that they form a triangle was previously calculated as 5/468.

Is this a coincidence, or a deep pattern?

I don't know, but they are mutually exclusive, so a 25-gon can be formed with probability 5/468 + 5/468 = 5/234