How can you predict the last card for a deck containing any number (n) of cards?

Question

Suppose you want to know the last card for a deck of 47 cards or one with 102 cards and you don't have that many cards.

What would be the equation I would use to solve this problem? I know the pattern looks like this...
The 1st column is the number of cards in the deck. The 2nd is the last numbered card.

1 = 1
2 = 2
3 = 2
4 = 4
5 = 2
6 = 4
7 = 6
8= 8
9 = 2
10 = 4
11 = 6
12 = 8
13 = 10
14 = 12
15 = 14
16 = 16
17 = 2
18 = 4
19 = 6




Please let me know how you solved it and please let me know what equation will work!

Answer 1

Well..... If there are n cards in the deck and it is shuffled honestly, you can predict what the last card will be. But, on average, you,ll only be right about 1 time in n ☺


Doug

Answer 2

in the above list, you can see that the number of card (n) = the last numbered card (f(n)) at 1,2,4,8,16 (or 2^0, 2^1, 2^2, 2^3, 2^4).

therefore the pattern will be

31 = 30
32 = 32
33 = 2
...
63 = 62
64 = 64
65 = 2
...
127 = 126
128 = 128

You can separate the series into sections
1 - 1 f(n) = (n-0.5) X 2 = 2n-1
2 - 2 f(n) = (n-1) X 2 = 2n-2
3 - 4 f(n) = (n-2) X 2 = 2n-4
5 - 8: f(n) = (n-4) X 2 = 2n-8
9 - 16: f(n) = (n-8) X 2 = 2n-16
17 - 32: f(n) = (n-16) X 2 = 2n-32
33 - 64: f(n) = (n-32) X 2 = 2n-64
65 - 128: f(n) = (n-64) X 2 = 2n-128

f(47) = 2(47)-64 = 30
f(102) = 2(102)-128 = 76