It seems like the chance of getting your Urchins to collide on turns 3-4 after a double Urchin opening should be a standard reference, but I don't see it anywhere (including the
Wiki page).
I calculate this as 5/11 = 45.5%
5/12 .
Method: Without loss of generality, consider the first Urchin. There's an 11/12 chance that it is not the bottom card of the shuffle (in which case it can't collide). If it's cards 1-5 or 7-11, then there are 5 places that the second Urchin could be so they collide (e.g., card 11 collides with cards 6-10, card 1 collides with cards 2-6, etc.).
Edit: There's a special case that I initially missed that dnkywin points out where if the first Urchin is in card 6, then 10 of the remaining 11 positions collide with it (all but the bottom card) instead of 5 for the other locations.
So the chance is (10/12) * 5/11 + 1/12 * 10/11= 5/11.
If you don't draw any Urchins on turn 3, then by the same logic as above you have a 70.4% (5/7) chance to collide the Urchins on turn 4 (6/7 * 5/6= 5/7). There is a 1.52% (2/12 * 1/11) chance that your Urchins are at the bottom of your shuffle, in which case you know on turn 4 that you will collide them on turn 5. Otherwise, if your Urchins don't collide you'll presumably want to add a third Urchin before the reshuffle to increase the odds of collision on the next shuffle.
Thoughts? Anyone confirm or disagree with the calculations?