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# A binary trick

By on 01/02/2016

The principle is simple:

Any integer can be represented as a sum of different powers of 2 in a unique way. For example, 21 = 16 + 4 + 1. This is called a binary representation of the number and is usually written as 10101 (1 time 16, 0 times 8, 1 time 4, 0 times 2, 1 time 1). Consequently, one can classify all positive integers into groups: those that have a 1 in their binary representation, those that have a 2, those that have a 4, etc. If the spectator indicates the correct groups, the performer can easily discover the number.

Say you want to guess a number smaller than a specific power of 2: 4, 8, 16, 32, 64, 128, … Then the cards corresponding to the groups with a 1 appearing, with a 2, with a 4, etc. would look like this:

For guessing a number smaller than 4:
Group with 1: 1, 3
Group with 2: 2, 3
I. e., there would be two cards with two numbers – the performer would need to ask twice if the number is on a card to find a number from 1 to 3 – boring, isn’t it?

For guessing a number smaller than 8:
Group with 1: 1, 3, 5, 7
Group with 2: 2, 3, 6, 7
Group with 4: 4, 5, 6, 7
I. e., there would be three cards with four numbers on each: the performer would need three questions of the type “do you see your number?” to find a number between 1 and 7. Slightly better than the above, but still not very impressive.

For guessing a number smaller than 16 we have:
Group with 1: 1, 3, 5, 7, 9, 11, 13, 15
Group with 2: 2, 3, 6, 7, 10, 11, 14, 15
Group with 4: 4, 5, 6, 7, 12, 13, 14, 15
Group with 8: 8, 9, 10, 11, 12, 13, 14, 15
This is already a so-so situation: four questions needed to discover a number between 1 and 15, and the cards have 8 numbers on each so also the principle is better hidden. It gets better for guessing numbers between 1 and 31, or between 1 and 63: there you need 5 or 6 cards with 16 or 32 numbers on each, so there is a reasonable amount of questions by the performer (5 or 6) which is significantly smaller than the number of possible guesses, and there are enough numbers on each card to camouflage the principle. Still, one should not overdo it: although in some cases “the more, the merrier” applies, here this is not a case: if one wants to guess larger numbers both the number of cards, i.e. questions, gets larger and thus boring, and the number of integers on them also increases, so that it becomes probable that the spectator will make a mistake in his answers.

Here are the cards for the 1-31 and for the 1-63 version:

Note that one can make the trick more attractive through individual styling, e.g. by asking the spectator to find his day of birth on the 1-31 cards and his birthmonth on the 1-15 cards and thus discovering his birthday.