# It's Not Magic, It's Mathematics

*I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.*

## Math Magic Tricks

Entertainers such as magicians and mentalists incorporate numbers into their staged illusions. I am referring not to sleight of hand card tricks or other such manipulations but a display of mathematics camouflaged by razzle-dazzle and cries of “abracadabra".

Although we know it is not real magic, it still seems like they are doing the impossible, just like creating impossible mathematics shapes such as the ones shown below.

## Understanding Number Magic

This article will hopefully go some way to demystify so-called number magic and encourage you to explore the fascinating world of number patterns and algebra.

## Magic 1: Is That a Zebra Crossing?

Let’s begin with one where I predict the outcome regardless of your initial choice of number.

Carry out these steps in turn, keeping track of your answer each time.

- Think of any number.
- Square it. That means multiply it with itself, such as 3 x 3, 8 x 8.
- Add the result to your original number.
- Divide the answer by your original number.
- Add 99.
- Subtract from the answer the number you started with.
- Divide by 10.
- Now add 16.
- If A = 1, B = 2, C = 3, D = 4, etc, work out the letter that corresponds to your final answer.
- Think of a 4-legged animal whose name starts with the letter you found.

I’m sure the animal you came up with has stripes and looks like a donkey!

Try this again using a different number. What can you conclude?

Now let’s see mathematically what is happening.

We will use the letter N to represent the start number and perform each of the 10 steps using this letter. The solution is shown alongside each step.

- Think of any number. [ N]
- Square it. [N x N]
- Add the result to your original number. [N + N x N]
- Divide the answer by your original number. [(N + N x N) ÷ N = 1 + N]
- Add 99. [100 + N]
- Subtract from the answer the number you started with. [100 + N – N = 100]
- Divide by 10. [100 ÷ 10 =10]
- Now add 16. [10 + 16 = 26]
- If A = 1, B = 2, C = 3, D = 4, etc, work out the letter that corresponds to your final answer. [26]
- Think of a 4-legged animal whose name starts with the letter you found.

[Zebra is the animal everyone would immediately come up with]

## Read More From Owlcation

We conclude that the number we start with has no effect on the final number, which is always 26.

## Magic 2: I Know Your Age

Here is one where you can precisely determine a person’s age even though their choice of the start number is completely random.

Let’s assume it is currently January 1, 2018, the person was born on 14/8/1995, and he chooses 4 as his start number. The solution is shown alongside each step.

- Ask them to think of a number from 2 to 9. [ 4]
- Multiply the result by 2. [ 8]
- Add 5 to the answer. [ 13]
- Now multiply by 50. [ 650]
- If the person has had their birthday, add 1767. If the person is yet to have their birthday, add 1768. [ 2418]
- Ask them to subtract from their answer the year they were born. [ 423]

The last 2 digits of the answer is their age. [ 23]

We can now show why this method works by letting N be the start number and writing down the result of each step in terms of N.

1. Ask them to think of a number from 2 to 10.

[ N]

2. Multiply the result by 2.

[ 2xN]

3. Add 5 to the answer.

[ 2xN + 5]

4. Now multiply by 50.

[ 100xN + 250]

5. If the person has had their birthday, add 1767.

[ 100xN + 2017]

If the person is yet to have their birthday, add 1768.

[ 100xN + 2018]

6. Ask them to subtract from their answer the year they were born.

[ 100xN + (2018 – year of birth)] or [ 100xN + (2017 – year of birth)]

100xN can only have the values 200, 300, …, 900. This can be ignored in the final answer. Then (2018 – year of birth) or (2017 – year of birth) is the person’s birth year, which is obtained from the last 2 digits of the answer.

## Magic 3: Hieroglyphics Prediction

This one is both interesting and easy to explain. We will use 46 as our initial number.

- Think of a number from 10 to 99. [ 46]
- Add its two digits together. [ 4 + 6 = 10]
- Subtract the total from the original number. [46 – 10 = 36]
- Find the shape next to your answer. [ a circle inside another circle]

It turns out that the answer will always correspond to a number with a circle next to it.

Let’s see why by reworking and explaining each step.

- Suppose our 2-digit number is AB. This can be written as 10xA + B. For example, 46 = 10x4 + 6.
- Add the two digits together to get A + B.
- To subtract the total from the original number, we write 10xA + B – (A + B).

This is the same as 10xA + B – A – B, which simplifies to 9xA.

Now, A is the first digit, which can be any of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9.

Therefore, 9xA are the first 9 multiples of 9.

Hence the only possible answers for choosing an initial number from 10 to 99 are 9, 18, 27, 36, 45, 54, 63, 72, 81 or 90.

If you look again at the diagram above, you will notice that the symbol next to each of these multiples of 9 is the same; a circle inside another circle.

## Magic 4: Symbols Galore

This one is an interesting variation of Magic 3.

- Choose two different digits and make a number from 10 to 99. Suppose we choose 5 and 7 to form the number 57.
- Reverse the two digits to get another number. 75
- Subtract the smaller number from the larger number. 75 – 57 = 18
- Find the symbol under your answer.

The shape is a box.

The following provides a proof that the result is always the same.

1. Suppose our two digits are A and B and we form the 2-digit number is AB.

This can be written as 10xA + B.

2. We reverse AB to get BA. This can be written as 10xB + A.

3. Let’s assume 10xA + B is the smaller of the two numbers.

Subtracting the smaller number from the larger number gives

(10xB + A) – (10xA + B)

This is the same as 10xB + A – 10xA – B.

This simplifies to 9B – 9A which is the same as 9x(B – A)

Now, the possible values for the difference, B – A, are 1, 2, 3, 4, 5,6 ,7 ,8, 9.

Therefore, 9x(B – A) are the first 9 multiples of 9.

Again, if you look at the diagram above, you will see that each multiple of 9 has a box shape adjacent to it.

As our final exploration, let’s look at an extension of Magic 3.

## Magic 5: It’s All Smiles and Smooth Sailing

- Pick any number between 100 and 999 with its first digit greater than its last digit. Suppose we choose 453.
- Reverse the digits and subtract the smaller answer from the larger answer. The reverse of 453 is 354. Subtracting 354 from 453 gives 99.
- Find your answer in the grid below.

A smiley face.

## Practice It Yourself

Do you think you can go solo in proving that the answer is always going to be a multiple of 99? Try it before looking at the solution given below.

Suppose our 3-digit number between 100 and 999 is ABC.

This can be written as 100xA + 10xB + C.

The reverse of ABC is CBA, which we can write as 100OC + 10xB + A.

Let’s assume 100xA + 10xB + C is the smaller of the two numbers.

Subtracting the smaller number from the larger number gives

(100xC + 10xB + A) – (100xA + 10xB + C).

This is the same as writing 100xC + 10xB + A – 100xA – 10xB – C, which simplifies to 99xC – 99xA. This can also be written as 99x(C – A).

The possible values for the difference, C – A, are 1, 2, 3, 4, 5,6 ,7 ,8, 9.

Therefore, 99x(C – A) are multiples of 99.

Examining the diagram above confirms that each multiple of 99 has a type of smiley face underneath it.

## Now You Know the Magician's Secrets

So, the next time you see a magician’s amazing number crunching or a mind-reader’s apparent probing of your mind, you will gently smile and say to yourself, “Yep, I know how it is done!”

## Comments

**Kari Poulsen** from Ohio on January 04, 2018:

These are really amazing. I'm going to have to memorize the first 2 so I can use them on my friends, lol.