Top hat revisited

Some years ago, I was talking about the top hat, an object that can be using for conjuring tricks and that, at low levels, allows you to improve your abilities in the skills of conjuring and charming (and, to a lesser degree, of binding and enchanting). But I notice that I didn't give any information about what I was able to conjure out of it, or what the bonuses required for each of the five elements that can come out of the top hat (hankie, ball, flags, dove, rabbit) are. I decided to check the Discworld Wiki on this, but apparently, there's data only for the first two items: 175 bonuses for the hankie, 250 for the ball. 

I've been checking what I can pull out at the moment. I succeed at everything up to the fourth item, dove. For the first two, I have no difficulties. For flags, I succeed about 70% of the time. Dove, on the contrary, is really, really tough: I might succeed about 30% or less. Rabbit is clearly impossible; I don't even make any progress at all.

When you try to pull something out of the hat, you invariably get what looks like... three or four checks:

You start to prepare yourself for the immense task ahead.
You flex your fingers over the black top hat and try to conjure up an image in your mind.
You picture xxx.

You wave your hands vigorously over the black top hat before delving deep in, then slowly removing your hand to reveal...

"You flex your fingers..." seems like it should check conjuring. I never fail this. "You picture..." doesn't seem clear as to what skills it is checking, but I can fail this, getting the message "You suddenly stare with dismay at your fingers".  I only fail at this stage with the dove, and that's all I reach with the rabbit.  Finally, one can fail the "You wave your hands..." check, in which case you get as output: "...absolutely nothing!". I can occasionally fail this with the flags, so I am guessing this is the hardest check. 

The outputs for successful magic read like this, for each item up to rabbit:

pull hankie from top hat
You start to prepare yourself for the immense task ahead.
You flex your fingers over the black top hat and try to conjure up an image in your mind.
You picture a silk handkerchief.
You wave your hands vigorously over the black top hat before delving deep in, then slowly removing your hand to reveal...
...an exquisitely embroidered aquamarine silk handkerchief.
You proceed to mop your brow with the hankie before disposing of it back in the black top hat.

pull ball from top hat
You start to prepare yourself for the immense task ahead.
You flex your fingers over the black top hat and try to conjure up an image in your mind.
You picture a billiard ball.
You wave your hands vigorously over the black top hat before delving deep in, then slowly removing your hand to reveal...
...a truly magnificent ivory billiard ball.
You throw the ball between your hands before hurling it high into the air and stylishly catching it again in the black top hat.

pull flags from top hat
You start to prepare yourself for the immense task ahead.
You flex your fingers over the black top hat and try to conjure up an image in your mind.
You picture the flags of all nations.
You wave your hands vigorously over the black top hat before delving deep in, then slowly removing your hand to reveal...
...the flags of all nations.
You run around for a while trailing the flags behind you in a procession of victory. You soon stop though, when you realise exactly how idiotic you look.
You stuff the flags back into the black top hat with an embarassed smile.

pull dove from top hat
You start to prepare yourself for the immense task ahead.
You flex your fingers over the black top hat and try to conjure up an image in your mind.
You picture a beautiful white dove.
You wave your hands vigorously over the black top hat before delving deep in, then slowly removing your hand to reveal...
...a beautiful white dove.
The dove looks around for a moment, flexes its wings, then flies into the sky. You watch it fly off until it becomes too small to see anymore.

My bonuses in the suspected skills are as follows: conjuring: 364, charming: 349, binding: 356, enchanting: 360. Charming is mental, so I can perhaps adjust it by not holding the klein bottle. Binding and enchanting are physical, so perhaps I can adjust them by checking results when not having the effects of Wungle's Great Sucking on me. Conjuring is spiritual.  

I am starting to think that maybe the top hat makes two skills checks only: the first one for conjuring, and the second for any/all of charming, binding and enchanting, with the first being much easier than the second.

Based on all this, I am guessing a conjuring bonus of c. 300 for flags and of c. 350 for the dove. Maybe a bit more for each? And something like 400 for the rabbit. For the other check, and assuming I have an average of 355 in the 3 skills involved (355.25 to be precise, weighting binding with 50% and each of the others with 25%) and the skillcheck being harder, I am guessing... a little bit more, but likely much more? As I fail the dove so much, it must be at least 360-70. So maybe something like c. 350 for flags, about 370 for dove, and likely over 400 for rabbit (420?).

I am interested in developing a research project here, so If you're interested, get in contact with me and we will explore this matter!