How the trick is performed:

Ask your guest to chose a number between 1 and 63 asking them not to reveal what the number is. The guest is presented with the 6 cards containing numbers. Ask them if their number is visible on each card or not. The host can predict accurately what the number is after going through the 6-card set once.

Psychology and misleading techniques:

Try to keep your guest busy with a cognitive subsidiary task. ("Think about a memory - try not to blink, visualise a garden, chose a colour" etc...) That will take the attention away from this mathematical equation trick. "Psychology trick" is a much more misleading path than pretending this is a "magic trick".

How to guess the number:

A mathematical explanation is granted below, but the simple steps to guess the number involve adding up all the first number on each card where the guest said "yes".

NB: make sure that the guest has the time to sufficiently read the card, so that their mistake does not make you end up with the incorrect number.

To take the attention away from you doing the sum of all the "yes" cards, you can maybe split the piles in 2, keeping all number cards visible. That way you can add the numbers without manipulating the cards too much (while diverting the attention to something else, if possible).

If you want to take the attention away completely from the numbers, you can memorise each card and its associated number (Mission Statement = 32 etc.). Though this increases the difficulty, it makes the trick all-the-more impressive. You can ask the guest to mix the cards with you never looking at the side with the numbers.

The mathematics behind the trick: BINOMIAL Numbers

About Binomial Numbers Binomial, or base-2, numbers are numbers based on powers of two, just as our decimal, or base-10, numbers are based on powers of ten. Recall that the decimal number 6351 really means 6×103+3×102+5×10+1. Binary numbers work in exactly the same way, except that they use powers of two rather than ten. So the binary number 10011 means 1×2 4 + 0×2 3 + 0×2 2 ++1×2 1 + 1×1, or 16+2+1 which is 19 as a decimal. As another example we find the decimal equivalent of 101011. This as a decimal is 1 × 2 5 + 1 × 2 3 + 1 × 2 + 1 × 1 = 32 + 8 + 2 + 1 = 43.

You might want to try 11011 and show that its decimal equivalent is 27. What does all this have to do with the magic trick? Take the number 19, as above, which is 10011 in binary. Remember that 10011 means 1 × 2 4 + 0 × 2 3 + 0 × 2 2 + +1 × 2 1 + 1 × 1 = 16 + 2 + 1. If we look at the cards with first numbers 16,2,and 1 we see that these three cards are the ones containing the number 19. That is, the number 19 has been placed on exactly those cards starting with the powers of 2 corresponding to a 1 in the binary expansion. (And so those numbers must sum to the number.) In the same way the number 27, which is 101011 in binary, or 1 × 2 5 + 1 × 2 3 + 1 × 2 + 1 × 1 = 32 + 8 + 2 + 1, and you can check that it is on the cards starting with 1,2,8, and 32. The same holds for any other number between 1 and 63.

Exercise:

Show that 1110 represents the number 14 and that 14 is on the cards beginning with 2, 4, and 8. Finally, we might ask how the cards were created. To do this we need to be able to find the binary representation of a decimal number. This is easily shown with a couple of examples. Recall that the first few powers of 2 are 1, 2, 4, 8, 16, and 32 . Number: 38. We start by writing 38 as the sum of powers of 2. The highest power of 2 that is less than 38 is 32, write 38=32+6. Since 6 is not a power of two, we find the biggest power of 2 less than 6, which is 4, so we now have 38=32+4+2. Since 2 is a power of two we are done. Adding in all the skipped powers of 2 gives, 38 = 1 × 32 + 0 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 0 × 1, or the binary number 100110. We would place the number 38 on the cards with first number 32, 4 and 2. Number: 15. Using the highest powers of two possible we find that 15=8+4+2+1, which is 1111 in binary. We would place the number 15 on those cards with first number 8,4,2, and 1. To make up a set of binary magic-cards we need to find the binary expansion of each number 1-63, and then place the numbers on those cards beginning with a power of two corresponding to a 1 in the binary expansion of the numbers.