Mathemagic: A way of motivating mathematical learning and having some fun in mathematics

Paul Swan, Edith Cowan University, Bunbury Campus 

Motivation

Sobel and Maletsky (1988) use the three Ds: Dull, Deadly and Destructive of all interest (p.2) to describe many mathematics lessons. In particular, children in middle school can become bored if mathematics lessons consist purely of ‘turn to page 24 and do all the even numbered questions’. Often children have a poor attitude towards mathematics and therefore don’t enjoy maths lessons. It should be acknowledged that while it may not be possible to stimulate and motivate each student in every lesson, it is important to engage students in the lesson. Resorting to ‘fun lessons’ or games afternoons, however, will not solve the problem. If fun is the focus of the lesson the children and the teacher might lose sight of the mathematics and sacrifice learning for the sake of smiles and laughter. There is a way to combine fun and mathematics content in a way that produces tangible mathematics outcomes - mathemagic!

Mathemagic should combine the mystery and intrigue of a magic trick with some mathematical content. Gardner (1956, p. xi) notes ‘mathemagical mathematics combines the beauty of mathematical structure with the entertainment value of a trick’. What better way to encourage children to do some mathematics in a non-threatening environment?

Most teachers have their own ‘bag of tricks’ made up of various challenges (e.g. How many pentominoes can you find?) stories (e.g. The story of young Gauss, grains of rice on a chessboard), and puzzles (e.g. The Tower of Hanoi) that they use to cajole their students into doing some mathematics. The purpose of this paper is to analyse some mathemagic, show how a piece of mathemagic can be developed in a mathematics lesson or an investigation - and to add to the ‘bag of tricks’.

Calendar magic

The calendar has an interesting history which in itself can be used to capture the imagination of children. The following piece of mathemagic relies on the structure of the calendar to work.

 Ask a volunteer to secretly draw a square around any four dates on the calendar (e.g. 7, 8, 14, 15). The volunteer is told to add all four dates and reveal the sum. The mathemagician then offers to reveal the original set of four dates.

How the trick works

To find the original set of four dates, divide by four and subtract four.

Where is the mathematics?

For younger students this trick provides a motivation to learn to mentally divide by four and subtract four. The division will never involve remainders so it should be within the grasp of most students. Older students can be encouraged to examine the algebra behind the trick. If the variable d is assigned to the first date, then the dates that follow are d + 1, d + 7 and d + 8. Adding these gives 4d + 16. When the total of the four dates is revealed, an equation is formed, 4d + 16 = 44. Removing a factor of four gives 4(d + 4) = 44. Dividing both sides by four produces d + 4 = 11. Subtracting four from each side of the equation leaves d = 7 or the starting date of the group of four. Adding one, seven and eight produces the set of four starting dates.

Brighter pupils can be encouraged to invent their own tricks based on the calendar. For example, what happens if a square is drawn around nine dates? Consider three dates in a diagonal or the corner dates in a square. Why restrict the investigation to squares? Consider parallelograms, rectangles and crosses. (Extensions to this problem may be found in Flexer, 1979 and Swan, 1994a.)

Mind reading cards

This trick is based on using the set of five cards below to determine a secret number. A volunteer is asked to choose a number between one and thirty-one. The five cards are then shown to the volunteer, one at a time, and the volunteer is asked to state which of the cards contains his/her number. Based on this knowledge the mathemagician can determine the secret number.

How the trick works

To work out the secret numbers, simply add each of the numbers appearing in the top left corner of each selected card. For example, if the volunteer said his/her number appeared on the first, third and fifth cards the secret number would be twenty-one (i.e. 1 + 4 + 16)

Where is the mathematics?

The cards are based on the powers of two. The number in the top left of each card is a power of two, i.e. and . By adding powers of two we can make any whole number. For example, fifteen can be made by adding one, two, four and eight, hence the number fifteen appears on the first four cards. Eleven, however, only appears on the cards with one, two and eight in the top left corner because these add to eleven.

For younger children the challenge is to mentally add the powers of two when the cards have been selected. Older students can be set the challenge to make their own set of cards. Brighter pupils might be challenged to extend the trick to include a sixth card with thirty-two shown on the top left. If a sixth card is added, all the cards will need to be modified to cater for numbers up to sixty-one.

Hansen (1992) describes how he makes and uses these cards with his classes. He goes on to describe a further extension to this activity involving the production of ‘Mind Probe Cards’ in base three. De Mestre (1994) also gives an excellent description of how this mathemagical trick may be used to advantage.

Seeing double

Ask your audience to follow these steps:

Write down any three single-digit numbers; e.g. 4, 6, 1.

Use the three numbers to make six two-digit numbers: 46, 41, 61, 64, 14 and 16.

Add the six two-digit numbers: 242.

Add the original numbers; 11 (i.e. 4 + 6 + 1).

Divide the larger number by the smaller, 242 ¸ 11 = 22.

Try this procedure again using a different set of starting numbers.

What do you notice?

How the trick works?

After trying this a few times the audience will soon realize that the answer will always be 22. This will certainly prompt the question why? If not, ask it yourself.

Where is the mathematics?

Time to introduce a little algebra. If we let a, b and c represent the three single-digit numbers, the six two-digit numbers are formed thus: 10a + b, 10a + c, 10b + a, 10b + c, 10c + a and 10c + b. Adding like terms we get 22a + 22b + 22c. Taking out a common factor of 22 produces 22(a + b + c). Dividing by (a + b + c) will therefore give 22 all the time.

Younger children can simply be given the task of determining whether the result will be twenty-two in all cases. There is only a finite number of examples to try. Answering the problem involves several additions and one division. Children rarely complain at having to complete all these calculations because they are keen to find a set of numbers for which the answer is not twenty-two. (Similar problems may be found in Swan (1993, 1994b.)

Count the cards

Prior to performing this trick you will need to prepare a set of eight cards. The first eight odd numbers are written on the front and the first eight even numbers on the back.

How the trick works

To work out the hidden number:

Find how many ‘even’ cards are showing. You are told this by the helper.

Add this to sixty-four.

In the above example the volunteer would let you know that there were four even numbers. To find the hidden numbers, simply add 64 + 4 = 68.

Where is the mathematics?

The reason this works is outlined below. If all the odd-numbered cards are added together (1 + 3 + 5 + 7 + 9
+ 11 + 13 + 15 = 64), a total of 64 is formed. Every time a card is flipped from odd to even, one more is added to the total.

While the above is not strictly a card trick, in the sense that it does not involve using a standard deck of playing cards, it is within the same genre. (Examples of card tricks based on mathematical principles may be found in Greenbury, 1994 and Morgan, 1998).

‘Think of a number’ tricks

‘Think of a number’ tricks may be found in most teachers’ ‘bag of tricks’. The teacher guides the students through a series of mental calculations that end up either with the starting number or a number stated at the beginning of the trick. At the very least these tricks provide children with the motivation to complete a series of mental calculations, but they may also be used to introduce some simple algebra. The link between the trick and algebra may be seen in the following example.

Once students have seen how this type of mathemagical trick is created they can try to design one of their own. This approach provides a simple introduction to algebra. Students can be challenged to modify the above example or create one of their own that ends up with a predetermined answer. For example, adding the instruction, ‘subtract the number you first thought of’ removes the original number and allows the mathemagician to introduce a predetermined finishing number anywhere in the set of instructions.

Figure 6. Think of a number, finishing with the number with which you started

The example below illustrates how a ‘Think of a number’ puzzle may be produced to finish on a predetermined answer - in this case, fifteen.

One way to start children thinking about puzzles of this type is to give them one that works (e.g. the one shown above) and ask them to alter the puzzle to finish on a different number. Altering line two or line four will affect the finishing number. Several authors (Flexer, 1979; Geer, 1992 and Morgan, 1995) refer to the use of these ‘Think of a number’ tricks in their classrooms. They also describe how the tricks may be improved with a few props like playing cards and some theatrics on the part of the teacher.

Magic numbers

Several authors (Sobel & Maletsky, 1975; Schaefer, 1981), describe several numbers that when multiplied appear to have magical properties. For example, the number 12 345 679 may be used to good effect. Simply ask a student to choose any single-digit number and then to multiply it by nine (a good way to get students to practise their nine-times table). Next the product is multiplied by 12 345 679. To add to the mystery of the occasion the mathemagician at this point can predict the outcome prior to the completion of the final calculation.

The following example shows the trick in operation. If a student volunteer chooses seven as the single-digit number, then sixty-three is produced as a result of multiplying seven by nine (7 ´ 9 = 63). The result of multiplying sixty-three by the magic number is 777 777 777. It should be noted that this calculation cannot be performed on an ordinary calculator with an eight-digit display so most students are suitably impressed when the teacher (mathematician) predicts the answer almost instantaneously.

How the trick works

The product of 12 345 679 and 9 is 111 111 111. Therefore any single-digit multiple of nine will produce the following result nnn nnn nnn, where n represents the single-digit multiple of nine.

Where is the mathematics?

The mathematics to be derived from a trick of this kind comes from the children developing their own magic number puzzles. Once the students appreciate the principle underlying this trick they may like to make up some of their own using one of the following magic numbers: 37 037, 15 873, 8 547. Development of the trick requires recognizing the relationship between these numbers and 111 111. (For a detailed discussion of magic numbers see Schaefer, 1981.)

Thrice dice

The addition of a few simple props such as dice can help maintain the illusion of a mathemagic trick. To complete this trick the mathemagician will need three dice; the larger the dice the better. A volunteer is asked to roll the dice and then to stack them on top of each other. The volunteer is then given the instruction to add the values of the five hidden faces of the dice (i.e. the face touching the table and the faces which touch each other). This total may be kept secret or written on a piece of paper and sealed in an envelope. The mathemagician simply glances at the number shown on the top face and offers to reveal the secret total.

How the trick works

The mathemagician subtracts the value shown on the top face from twenty-one.

Where is the mathematics?

The mathematics lies in the fact that the opposite sides on a fair die add to seven. The total of the three dice will always be twenty-one. Children are often really impressed if you stack ten dice on top of one another. In this case the value shown on the uppermost face is subtracted from seventy.

Once the mathematics behind the trick is explained, students can be encouraged to number nets of a cube so that they fold to form a fair die (i.e. one where opposite faces add to seven). Better still the children could be encouraged to find arrangements of six squares that produce nets for cubes. There are eleven different nets that may be folded to form a cube. Once several nets have been found, the children could be encouraged to explore the question. ‘Are some nets for cubes (dice) easier to number than others?’

Multiplication shortcuts

Children love to learn shortcuts to help them save time when multiplying. Often children are taught the following trick for squaring numbers that end in five.

To square a two-digit number that ends in five (e.g. 35):

The last two digits will always be twenty-five.

The left-hand digit(s) are found using this procedure:

Square the tens digit 3 ´ 3 = 9.

Add the tens digit to that product: 3 + 9 = 12.

The answer in this case is 1225.

The procedure works for any two-digit number ending in five, but the real mathematics appears when the question of why it works is posed.

How the trick works

This multiplication shortcut leads nicely into the following piece of algebra. (not as many students will suggest)

Using the above example,

Where is the mathematics?

Calculation shortcuts may be found in many places (see, for instance Schaefer, 1981; Swan, 1993, 1994b and http://forum.swarthmore.edu/k12/mathtips.html). Older students could be encouraged to look for other multiplication shortcuts. Younger children will simply be keen to impress their parents with their lightning abilities to calculate.

Marvellous Mobius

Not all mathemagic relies on algebra, pattern or number to work. The Mobius band has been included as an example of a topological trick. Certain knot-tying tricks work on similar principles. To perform this trick will require the production of a few props and some scissors.

A Mobius band is produced by joining a long strip of paper (the longer the better) after twisting the strip half a turn. A volunteer is asked to cut the strip lengthways to produce two thinner strips. The audience is always fascinated to find that two interlocking strips are formed instead of two separate strips. Challenging the volunteer to cut the strips lengthways again produces an even more fascinating result.

How the trick works

When the strip is made, one surface is effectively created by placing the half-twist in the strip. Effectively there is no longer an inside and outside, just one continuous surface. Cutting the strip will no longer cause the strip to separate into two halves.

Where is the mathematics?

Children of all ages and abilities can explore what happens if the resultant strip is cut again and again, or what will happen if the original strip was made with a full twist. Work with the Mobius strip relates to other ideas in topology. (Several practical applications of the Mobius strip are discussed in Jacobs, 1970, and Pappas, 1989, 1991).

Conclusion

The above activities all have an element of fun for the students involved but the fun has to be linked to the development of some meaningful mathematics. Teachers are always under pressure to complete a body of content within a particular time frame so it is important that any activity be linked to meaningful content. Teachers and students must be clear about the mathematics underpinning the activity, otherwise opportunities to capitalize on the motivation provided by the mathemagic will be lost.

For those wishing to expand their repertoire of mathemagic the following list of references and websites will be of use.

Books

Websites

The following site contains sections on calculational wizardry, geometry and topology based mathemagic and puzzles, and some card tricks:

http://www.scri.fsu.edu/~dennisl/CMS/activity/MM.html

The Math Forum at Swarthmore contains a great deal of useful information. Of particular interest to mathemagicians is the section ‘calculation tips and tricks’ which contains a host of calculation shortcuts. The site may be found at:

http://forum.swarthmore.edu/k12/mathtips.html

Cyntia Lanius maintains an interesting site which has links to some fascinating material on calendars. Her site may be found at:

http://math.rice.edu/~lanius

References

Footnote:

This paper is reprinted with permission of the Mathematical Association of Victoria as it was originally published in the 1998 Conference Proceedings.