Mathemagic

Two Squares

This trick appears to be mathematical but perhaps it is not!

Can you change this square into two using only a pencil? You are not allowed to come anywhere near the screen (or whiteboard).

Square

 

The Trick: Carefully use the blunt end of a pencil to push up the lower eye lid of one eye and you will see double!

Coming Together

Each student is asked to think of a number between 2 and 9 inclusive (or ace, jack, queen or king). Whatever number they have chosen, there are four cards in their pack with that value. Students are asked to shuffle the pack while thinking about their cards.

Playing Cards

Students are considered to have magical powers if two of their cards come together in the pack (well there might be one card between them but for the purposes of this exercise we will call that coming together).

Most students should find that this works for them ... so they have magic powers ... or have they?

The Trick: In fact this is an example where the odds are weighted in favour of a success ... can you estimate the probability from the results of this activity?

Think of a number

"There aren't many grey elephants living in Denmark are there?"

Virtually everyone in the room will have thought of Denmark as their country and as elephant as their animal. Is this magic?

Students can now investigate the probability of this happening. Having first satisfied themselves that everyone will arrive at the letter D independently of their starting number, how many countries are there starting with D?

How many animals beginning with E? Of all the animals beginning with E how likely is a person to choose elephant? Perhaps a survey might answer this question.

Students could now be asked to make up for themselves a similar trick (but not using Elephants from Denmark). Try out the tricks on some other students and teachers.

Finally a demonstration of tree diagrams can be used to explain the probabilities of these tricks working.

Trick Tree

Update November 2014: The 96% and 85% on the tree diagram above were made-up figures for illustrative purposes only. We have now conducted our own survey to see what those figures should be. You can see the results on the Predictive Survey Results page.]

Amazing Adding

You can amaze your students by pretending to add up a whole column of three digit numbers in a couple of seconds.

Ask three students to write three digits each on a piece of paper as if they are to be added and return the paper to you.

You write two more numbers below those. Choose your first number (which is the fourth one in the problem) so that the first number and fourth numbers add up to 999. Choose your next number so that it adds with the second number to give 999.

283
234
549
716
765

Give a student the paper and ask them to add up the numbers on a calculator. Then the student could return the paper to you. Look at it briefly as if you are adding the numbers in your head and announce the answer.

The answer will be 2000 plus the third number minus 2 which you could compute very quickly once the student returns the paper.

This could be followed by a class discussion about how this trick works ... students practise with friends.

Letters In A Number

The Teacher writes a number on a large piece of paper. The students can see this going on but can't see which number it is. The paper is the put in an envelope or a box and put in a safe place in view of the students.

The Teacher then asks a student to think of a very large number. This number is written on the board in words.

"How many letters are in these words?" ... the answer is written underneath in words...."How many letters are in these words?" ... and so on repeating the process again and again till the list of numbers has reached the bottom of the board. The envelope is then opened to reveal the number FOUR which will be the last number written on the board.

F O U R

"What if we start with a different number?" ... another student volunteers a different starting number which eventually generates four again.

"Does this always happen?".. students try some of their own.

"Why does it always happen?"

"What about other languages?"

Students now do an investigation looking at the properties of the number words in different languages. They can use foreign language dictionaries or the internet as appropriate. They should produce a write up of their findings for their folders.

Mental Tricks

many calculations have short cuts or tricks associated with them which enable students to perform the calculations quickly in their heads.

A list of such tricks can be found here.

Here are a couple of examples:

Quick Square
Did you know that there is a quick way of squaring a two digit number which ends in 5?
Just multiply the first digit by that number plus one.... stick a 25 after your product and there's your answer .... simple ?

Example:

Q. What is 35 squared?
A. 3x4=12
.....now stick on the 25
...........the answer is 1225

Do you know why it works?

Times Eleven
The eleven times table has always been very easy to learn up to 9 x 11. Here's a way of multiplying large numbers by 11 too:

Example:

Q. What is 324 x 11 ?
A. Write down the first digit ... 3
.......Add the first and second digits ... 3 + 2 = 5
.......Add the second and third digits .. 2 + 4 = 6
.......Write down the last digit ........... 4

The answer is 3564.

Try it yourself ... multiply 543 by 11

Do you know why it works?

Does it work for all three digit numbers?

Cluedo

A large house plan made from 9 pieces of carpet (each 3m by 3m) is set out in the middle of a large room as shown:

Cluedo

A student is dressed to look like the murderer(!) and stands on the carpet piece marked "Hall".

A teacher (acting as Sherlock Holmes) explains that the murderer will be tracked down in the house and arrested. Sherlock is blindfolded.

A second student throws the dice and states the number on the dice out loud. The murderer walks through that many doors on the plan of the house.

Sherlock says "I know the murderer is not in the kitchen (or name of other room) , so will someone please take away the kitchen"

A third student takes away the piece of carpet labelled "kitchen". the murder from now on cannot step on the area which was once the kitchen.

The process is repeated a number of times until Sherlock can say "I know that the murderer is in the ........... (name of room), you are under arrest.

"How does the trick work?"

Well if the murderer starts in the hall, four rooms are one door away (study, bathroom, dining room and conservatory). We shall call these rooms odd rooms.

The other rooms are two doors away. We'll call these rooms even rooms.

If the number on the dice is even, Sherlock knows that the murderer must have walked into an even room so can ask for one of the odd rooms to be removed.

The process continues. Sherlock uses the principles that:

Odd + odd = even

Even + even = even

Even + odd = odd

Odd + even = odd

... to keep track of the rooms the murderer can't be in. By process of elimination the correct room can be found (In practice Sherlock must reserve the right not to remove a room in certain cases)



This could be followed by whole group discussion about how the trick was performed (using student's own theories and words.

Students in pairs try to perform the trick using the a diagram and a counter to represent the murderer.

Acting as Sherlock requires a great mental effort keeping track of the rooms.

Memory Tricks

A number of items can be memorised in order by associating each item with a place on a familiar journey.

For example 20 geometric shapes could be visualised at various places on the journey from bed to school each morning. The shapes could be visualised at places such as the bathroom, kitchen bus stop etc.

Students challenge each other to memorise the order and names of 20 shapes such as pyramid, sphere, octagon etc.

One technique for remembering a list of items in order is by associating each item with an item from the following children's rhyme:

One is bun
Two is shoe
Three is tree
Four is door
Five is hive
Six is sticks
Seven is heaven
Eight is gate
Nine is line
Ten is pen etc.

This activity could start by collecting from the students a list of ten popular video films. Students are then asked to conjure up in their minds a crazy image of the item from the list above involved in the action of the film.

Eg if the fifth film is The Sound of Music; a student might picture Julie Andrews being chased over the Austrian Mountains by a swarm of angry bees.

Working in pairs students could practice this technique by memorising ten items from a shopping list

Mind Reading

Many mind reading tricks involve a "plant" in the audience secretly giving information to the mind reader. The way this information is given often involves mathematical codes or systems.

Nine objects are arranged in a 3 by 3 grid formation

Magic 3 by 3

While the magician isn't looking, the students are asked to choose an object an concentrate on it. The magician can quickly identify the chosen object by feeling the thought vibrations coming from it ... well not really. One of the students has previously been asked to act as the "plant" giving the magician a secret sign. Students are asked to work out some secret signs which could correctly identify one of the nine objects.

One system could be that the plant and the magician agree to define each object in terms of its row and column (like a matrix). The plant uses the position of their head to indicate the row and arms to indicate the column.

Students are encouraged to invent similar (more complex) illusions.

(alternatively this trick could be performed using nine buckets placed upside down on the floor with an item being put under one while the magician isn't looking.)

"Black Magic" is a well known old party trick in which a person apparently mind reads to find an object previously selected by an audience. An accomplice helps by pointing to various objects asking "is this it?". The mind reader always says no till after the accomplice has pointed to a black object; the next object to be pointed out will be the chosen object.
Students could devise similar mind reading tricks using number codes of other systems.

Numbered Cards

One person thinks of a number 1 - 20. They are asked which cards contain their number. The magician can work out the number by adding up the top left numbers on each of the selected cards.

Magic Card

How does it work? The numbers in the top left of each card are powers of two and any number can be expressed as a combination of powers of two. this can be shown to be the basis of the binary system.

This trick could be described by a teacher without showing the students a set of cards. Students challenged to make their own set of cards and extend for numbers greater than 20 (the trick works better the more numbers there are on each card). Students should practice performing this trick on each other using their own set of cards

Here's a more detailed Interactive version
The Online Psychic

Real Magic Square

Wouldn't it be impressive if you could instantly write down a 4x4 magic square giving any total that a friend suggests. It can be done ... there's a trick to it!

First you have to memorise the basic 4x4 magic square which gives a total of 34. (Memorising this could lead to some interesting multisensory work!)

14 1 12 7
11 8 13 2
5 10 3 16
4 15 6 9

Let's suppose your friend chooses the number 57. Firstly subtract 33 from this giving 24.

Now replace the numbers 1, 2, 3 and 4 in the basic square with 24, 25, 26 and 27 in that order.

You now have a magic square with 57 as the sum of each column, each row, each diagonal and each 2 by 2 square within the larger square.

Why does this work? Look at the positions of the numbers 1 - 4. How are they significant?


Your Comments

"This has saved me a lot of time over my summer holidays - thank you."

D. Boyce, Rohwell
Monday, August 26, 2013

 

"Thank you and blessings."

Sadhana Bhat, Mpsmkbh@gmail.com
Saturday, October 19, 2013

 

Do you have any thoughts about the mathemagic above? Do you know any mathemagical tricks that we could share with the world on this page? Click on the link below to submit your ideas.

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