# Algebraic magic

8th October 2004 at 01:00
Q How do you get a middle set Year 10 a bit interested in algebra?

A Bring this up at a department meeting to find out how your colleagues make algebra interesting. Recently, I ran a four-day event, Maths on the Menu, with Professor Chris Budd from Bath University. We discussed this very question and he directed me to his book, Mathematics Galore. He pointed out that many "magic" tricks have their base in algebra. Use one as a lesson starter - the nice thing about using algebra for magic is that the activity can be adapted to suit the class. I recommend the following one for this class.

You will need a clock face - it could be just a picture - with numbers on it. Tell the class you are going to read someone's mind. Begin by asking a pupil to choose a number on the clock face and secretly write it on a piece of paper. Explain that you are going to tap numbers randomly on the clock face, and that at the same time they should start counting (in their head) from their number, shouting "stop" when they have reached 20.

Although you say you are going to tap the numbers randomly, in fact you only tap some randomly - in this case, eight of them. The random tapping is important to fox the class. The next taps are in order: the ninth is on 12, the tenth on 11 and so on anticlockwise round the clock until the 20th.

Assume that the pupil chose 12. They count 12, 13, 14, 15, 16, 17, 18, 19 for the first eight taps. The ninth tap is on 12, which is on their count of 20, and Hey presto - as if by magic, your finger will have landed on the right number. If they had chosen 11, the ninth tap would have taken them to 19. But when you tap the 10th time, they get to 20 and you land on 11.

I've put the possible results in a table. The blue numbers at the top show the first eight random numbers and then the anticlockwise progression. The black numbers show the counts the pupil makes from the chosen number (first row 12, then 11, and so on).

The numbers highlighted in green show the position of the tap during the non-random anticlockwise progression. We can see from the table that the number of non-random taps plus the chosen number always adds up to 13.

Look at the first number highlighted in green. This shows that there has been one non-random tap, and the number is 12. The next green row down shows two numbers, the last being the number the turn finishes on: 11; with 11 + 2 =13.

To put this in algebraic terms: let x be the number the player thought of and * be the number of non-random taps. In each case, we should find in the table that x + * = 13. So if nine is the chosen number (x = 9), you can see that there would be four non-random taps (z = 4), and 9 + 4 = 13. The key is algebraic: the mathematical identity y - (y - xx where x is the chosen number and y is 13 (the largest number on the clock face + 1). By understanding how it works, pupils can adapt it and come up with their own magic tricks.

There are two parts to this trick. The one is shown in the first table: the important result is that the number of non-random taps + the chosen number always makes 13 (* + x = 13).

We use this below in a different form (* = 13 - x) to find out how we get y to be 13. Consider a ridiculous example, whereby we ask the pupil to shout out when they reach 12, and assume 12 is the chosen number. As we tap the clock face for the first time they will shout out. In this case, we have to start tapping on 12 to make the trick work - no random taps. But we have still tapped the clock once, not zero times.

Look at this example another way:let w be the number we land on after tapping the clock face non-randomly (here 12) and * be the number of non-random taps (here 1). Working from 12, we tap the clock face in order and put the results for w and z in a table.

In each case the number of taps plus the number landed on is 13: w+* =13.

Or, w=13-* . This is where the 13 comes from.

Now put the two parts together. From the first table we obtained the equation * =13-x. From the second part we got w=13-* . We can now replace * in the second equation using its value from the first: w = 13-(13-x). Which gives w=13-13+x. The two 13s cancel each other to leave w=x.

All that a magician has to do now is decide how many random hits they want before they start tapping in order, and so to what number a player must count to before shouting stop. In this example, there where eight random taps and 12 numbers on the clock face. These are added together to get the number the player has to count to: 20 in this example.

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.

www.nesta.org.uk

Email your questions to Mathagony Aunt at teacher@tes.co.uk Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

Other maths magic tricks can be found at http:motivate.maths.orgconferencesconf21c21_references.shtml

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