Regarding the shuffle, the simple cases can be reasoned about with math pretty directly, but adding in the x2 and especially the reveal both change the dynamic a lot. I wrote an app to simulate this to see which combination of rules is actually the most beneficial and by how much:
----: Grand: 1000000 Daily: 62500
---R: Grand: 1000000 Daily: 62500
--D-: Grand: 1000000 Daily: 117463
--DR: Grand: 1000000 Daily: 131502
-S--: Grand: 1133275 Daily: 66801
-S-R: Grand: 1142910 Daily: 81208
T---: Grand: 1058618 Daily: 66373
T--R: Grand: 1060055 Daily: 66819
T-D-: Grand: 1058975 Daily: 125130
T-DR: Grand: 1066201 Daily: 140102
TS--: Grand: 1209772 Daily: 71280
TS-R: Grand: 1222986 Daily: 87924 <-- This is what the event actually uses
T = 'Two X'; S = Shuffle; D = Daily Shuffle (winning the daily prize automatically shuffles the board); and R = Reveal.
The reveal strategy is (in order)
* If the 2x is visible and you haven't drawn the daily, draw the 2x.
* If the daily is visible, then pick the daily.
* If the shuffle is visible and you've already found the daily, draw the shuffle.
The numbers are for a million 'picks' from the board. 'Grand' is the number of 'grand prize' tokens you get (normally it's 1:1 but both 2x and reshuffle give you a free pick).
It's trivial to see where some of the numbers come from: If there is no reshuffle at all, you'll gather exactly 62,500 prizes (one every 16 picks). If you only have the reshuffle, because that's basically a free square, you'd average one prize for every 15 picks, which is what the numbers show.
Making the board automatically reshuffle when you draw the daily prize (which is virtually equivalent to having a separate 'reshuffle' button assuming you'd only press it after you hit the daily) almost doubles your rate of return.
Aside from that rule change, note that all of the other rules help you out. The rate of 87924 per million picks is about one daily prize every 11.4 plays. Without a way to reshuffle, you're always drawing all 16 presents before moving on. Reveal helps you tactically grab dailies or reshuffles or helps you avoid junk. And it's obvious why 2x is a good thing.