Christmas Cracker: Mystery Calculator Cards...

For once my Christmas cracker, cracked at a home party, contained something other than the usual useless Fabergé egg, so fashionable down our way. No, I got a pack of six cards, the so-called Mystery Calculator, which provides a neat little party trick for the non-initiated.

It goes like this. The six cards all have 32 numbers on them, all between 1 and 63 and arranged in a matrix of 4 rows and 8 columns per card. The unsuspecting participant is asked to randomly pick one card from the deck of 6 and to silently memorise one of the 32 numbers on that card. The card is then handed back to the magician and the participant is asked to select from the 5 remaining cards those that also have the memorised number on them.

The magician, based on the initially chosen card as well as the cards selected in the second step, is then able to divine which was the number the participant had silently chosen in the first place. With a bit typical magician's hocus pocus and mumbo jumbo, this neat little trick will draw some ooohs! and aaahs! from the party audience.

So what's the math behind the magic? All the magician has to do is to add the left hand corner numbers of the initial card and the other selected cards and this is always the number the participant had silently memorised!

It works like this. Although the numbers on the cards appear randomly selected numbers between 1 and 63, they're actually chosen specifically to appear on each card.

Each integral number can be broken down as a sum of powers of two, for example 15 can be written as 2³ + 2² + 2¹ + 2⁰ = 8 + 4 + 2 + 1. The binary form of the number 15 is thus 1111. Similarly, the binary code of 33 = 2⁴ + 2⁰ = 32 + 1. The binary code of 33 is thus 10001.

The numbers in the left hand corner of the six cards are 1 (2⁰), 2 (2¹), 4 (2²), 8 (2³), 16 (2⁴) and 32 (2⁵). To determine which other 31 numbers between 1 and 63 have to appear on each card, arrange them all in a digital matrix as follows:

1 = 00001
2 = 00010
3 = 00011
4 = 00100
5 = 00101
...
15 = 01111
...
33 = 10001
34 = 10010
...
63 = 11111

The numbers which have to appear on the card with 1 in the left hand corner are the ones whose digital representation contains a 1 in the last column, i.e. (1,) 3, 5, 7,..., 63.

The numbers which have to appear on the card with 2 in the left hand corner are the ones whose digital representation contains a 1 in the one before last column, i.e. (2,) 3, 6, 7, 10, 11, 14, ..., 63.

The numbers which have to appear on the card with 4 in the left hand corner are the ones whose digital representation contains a 1 in the column 2 places from the last column, i.e. (4,) 5, 6, 7, 12, 13, ..., 63.

And so forth. Try it, it works every time. Here are the cards and the explanation.

An alternative version of the trick consists in asking the participant to simply silently choose any number between 1 and 63 and memorise it. Then ask him to select from the magical cards those who contain the memorised number, do your hocus pocus (involve a 'beautiful assistant', if you've got one at hand), add up the left hand corner numbers of the selected cards and you've got the memorised number...