In this trick, you will give a student six cards of numbers
and then have her pick a number from 1 to 63.
Instruct her to tell you on which cards her number appears,
and you will tell her the number she picked!

Here are the six cards:

1 3 5 7 9 11 13 15
17 19 21 23 25 27 29 31
33 35 37 39 41 43 45 47
49 51 53 55 57 59 61 63
2 3 6 7 10 11 14 15
18 19 22 23 26 27 30 31
34 35 38 39 42 43 46 47
50 51 54 55 58 59 62 63
4 5 6 7 12 13 14 15
20 21 22 23 28 29 30 31
36 37 38 39 44 45 46 47
52 53 54 55 60 61 62 63
8 9 10 11 12 13 14 15
24 25 26 27 28 29 30 31
40 41 42 43 44 45 46 47
56 57 58 59 60 61 62 63
16 17 18 19 20 21 22 23
24 25 26 27 28 29 30 31
48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63
32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47
48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63

When having your student tell you the cards,
have her identify the cards by the number in the top left corner.
This will make it easier for you to tell her the number that she picked.
All you do is mentally add up the numbers that she gives you!

Why does this work?
The cards are based on binary arithmetic (base 2), so I used this trick whenever I taught number bases (like binary and hexadecimal and even our decimal system).   I use this as a teachable moment -- the students always want to know if I really memorized all the cards!

Recall the decimal system (base 10):   Look at the columns, going from right to left.   The first column is the units column.   The second column is the 10s column.   The third column is the 100s column.   Each succeeding column is multiplied by 10.

In base 2, the columns are multiplied by 2.   The first column is the units column, the next is the 2s column, the next is the 4s column, the next is the 8s, the next is the 16s, the next is the 32s, etc.   Did you notice that the first digit on the six cards corresponds to the first six column headings in base 2?   They are 1, 2, 4, 8, 16, and 32.

Now think how a number is represented in base 2.   Take the number 13.   In base 2, only 1s and 0s are permitted, so the number 13 is made up of one 8, one 4, zero 2s, and one 1.   It is written 1101.   Now look at the cards: the number 13 appears on the cards beginning with 1, 4, and 8!

I like to think of the cards as just the column headings for base 2.   I visualize them as a table.
  Here is a table with several examples:

Number
Picked 		Light Blue Card
32 		Light Purple Card
16 		Magenta Card
8 		Orange Card
4 		Yellow Card
2 		Light Green Card
1
3 = 0 0 0 0 1 1
13 = 0 0 1 1 0 1
16 = 0 1 0 0 0 0
32 = 1 0 0 0 0 0
42 = 1 0 1 0 1 0
63 = 1 1 1 1 1 1

P.S. I found these six cards in a box of Trix Cereal!