Obtaining ordering of cards
Python
def game(a):
    deck = range(1,a+1)
    #store the deck
    origDeck = []
    #copy the deck
    origDeck.extend(deck)
    table =[]
    ans = 0
    while 1:
        while len(deck) &gt; 0:
            table.append(deck.pop(0))
            if len(deck) == 0:
                break
            deck.append(deck.pop(0))
        table.reverse()
        deck,table=table,deck
        ans+=1
        #this is the condition that exits the infinite loop
        if deck==origDeck:
            break
    print ans

Calling game(n) prints the number of rounds for n cards. The code can be easily changed to allow for printing the deck between rounds (add print deck before if deck==origDeck).

Tests (1 through 50):
1: 1
2: 2
3: 3
4: 2
5: 5
6: 6
7: 5
8: 4
9: 6
10: 6
11: 15
12: 12
13: 12
14: 30
15: 15
16: 4
17: 17
18: 18
19: 10
20: 20
21: 21
22: 14
23: 24
24: 90
25: 63
26: 26
27: 27
28: 18
29: 66
30: 12
31: 210
32: 12
33: 33
34: 90
35: 35
36: 30
37: 110
38: 120
39: 120
40: 26
41: 41
42: 42
43: 105
44: 30
45: 45
46: 30
47: 60
48: 48
49: 120
50: 50

31 produces a peculiar output; it goes through 210 rounds.
It appears that primes have large cycles. For example, 97 has the largest cycle under 100: 6435.