# Count On

## Maths and Magic

### Maths & Magic

#### A Real Mystery?

Have a look at this "Mystery Calculator" which came out of a cracker this Christmas.

The complete set consists of 6 cards printed with a series of numbers. Show all the cards to a friend and ask him of her to select one number from any one card. Show the other five cards to your friend asking him or her to say whether the number appears on these cards. Take all the cards on which your friend says the number appears, add together the top left hand corner number of each card and the total is the number your friend selected.

How does this work? Well, take a look at those top left hand corner numbers. You will find them to be: 1, 2, 4, 8, 16, 32. There should be something undeniably familiar about this set of numbers - they are all powers of two: 2^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}. I expect you can begin to see how the calculator works now. All the whole numbers can be written in terms of power of two, using each power once or not at all. For instance, 23 is 16+4+2+1. Try it for all the numbers up to twenty.

On each of the cards you will find that the numbers listed are those which contain a certain power of two when you write them like this, and that power is the number in the top left hand corner. Obviously no two numbers can have the same combination of powers, so you can be sure that no two numbers can appear on the same combination of cards. Your friend will pick out for you this unique combination of cards and you will know that the chosen number contains those particular powers of two. So all you have to do is add them up! Simple!