# Mathemagic! Magic with numbers

By Sam Wheeler Friday, September 26, 2014

Hello and welcome to the final edition of Mathemagic. If you joined us last week you may have been trying to solve the puzzle and some of you may have solved it yourselves, check out the solution in the first trick.

## Number Divination

1) Ask your friend to write down any three digit number without showing you. Make sure the first and last digits are different to each other (e.g. 582).

2) Ask them to write down their number backwards (e.g. 285)

3) Now ask your friend to subtract the smaller number from the larger number, they should keep the answer a secret (e.g. 582 – 285 = 297).

4) Finally ask your friend to tell you either the first digit or last digit from their new number.

5) You will now tell them the other two digits!

How? I hear you ask! Well, firstly the middle digit will *always* be 9 – that is the easy part. Secondly, the first and last digit will always add up to 9. Whatever number they say you just subtract from 9 and you instantly know the other digit:

• If they say **“0” *then the other digit must be a *“9”**.

• If they say **“1”** then the other digit must be an **“8”.**

• If they say **“2”** then the other digit must be a **“7”**.

• If they say **“3”** then the other digit must be a **“6”**.

• If they say **“4”** then the other digit must be a **“5”**

.
• If they say **“5”** then the other digit must be a **“4”**.

• If they say **“6”** then the other digit must be a **“3”**.

• If they say **“7”** then the other digit must be a **“2”**.

• If they say **“8”** then the other digit must be a **“1”**.

• If they say **“9”** then the other digit must be a **“0”**.

## Why does this work?

When you reverse the number and calculate the difference, the middle digit will always be a 9 because you subtract the middle digit from itself (getting 0), but then you will also be subtracting the larger last digit from a smaller digit which will result in a negative number. That will then be subtracted from the middle digit which will in turn subtract from the first digit. Best to explain with an example: 582 – 285

So start by subtracting the hundreds, so 500 – 200 = 300

Then subtract the tens, so 80 – 80 = 0

Finally subtract the units, so 2 – 5 = -3

So now we add all of those together, so 300 + 0 + -3 = 300 – 3 = 297

You don’t need to understand why it works, you just need to remember how to do it!

Have a go at this one yourself and let’s see if I can read your mind!

1) Write down any three digit number. Make sure the first and last digits are different to each other (e.g. 582).

2) Write down their number backwards (e.g. 285)

3) Now subtract the smaller number from the larger number (e.g. 482 – 284 = 297).

4) Now reverse your new number (e.g. 297 reversed is 792)

5) Using your new numbers add the smaller number to the larger number (e.g. 792 + 297 = ....)

6) Concentrate on this total...

7) You are now thinking of the number 1089!

This mathematical technique will always result in the number 1089.

## Why does this work?

This is an extension of the Number Divination, but we will use the same method to force your friend to think of a specific number. So following on from the Number Divination, once you have got to the end of step 3 you will have a number where the first and last digits add up to 9. So if you reverse that number and add the first digit of the reversed number to the first digit of the non-reversed number you must get 9, and the same is true for the last digits. The middle digits of both numbers will always be 9, so they will add together and result in 18. So if you split all of these sums into columns you will have 9 in the hundreds, 18 in the tens and 9 in the units. Since 18 isn’t a single digit the 1 will go over into the hundreds column, so 9 + 1 = 10 and the 8 and 9 will stay in the tens and units columns resulting in 1089!

## How to use this:

You have a few options for how to use this number... • You could write down a secret prediction (1089) and ask your friend to place it in their pocket. Then go through the process and when you get to the end you ask them to tell everyone their number, now you ask them to reveal the prediction.

• You present this as mind reading. When you get to the end of the process you ask your friend to really concentrate on the first digit, and you name it. Then you proceed to name the other 3 digits one at a time. You can make this look really easy or really difficult by either naming the number instantly or pretending to really struggle to get the number they are thinking of.

• You could also use this to in more creative ways. For instance if you or your friend has a book (or maybe there is a book nearby), flick through it and secretly turn to page 108 and remember the 9th word on the page. Now do the number force and instead of revealing the number ask them if they have a 3 or 4 digit number, they will say “4”. Ask them to take the first 3 digits and to turn to that page in the book, and then to look at the last digit and to remember the word in that position on the page – they will turn to page 108 and remember the 9th word. Hopefully if the word is an interesting word like “tractor” you can have fun revealing the word they are thinking of – ask them to picture it and then you can begin describing it, but if the word is less interesting (like “and”) then get them to picture the word written big and bold in front of them, stare into their eyes for a while and eventually reveal the word they are thinking of.

## The Missing Number

1) Ask your friend to take out their calculator, or open the calculator app on their phone and then to think of any 2 digit number (e.g. 59).

2) Ask them to add both of the digits together (e.g. 5 + 9 = 14)

3) Now ask them to subtract the answer from their original number (e.g. 59 – 14 = 45)

4) Ask your friend to multiple their new number by any other 2 digit number they like (e.g. 45 x 98 = 4410).

5) Tell them to continue multiplying by 2 digit numbers until they get to a number which is between 7 and 8 digits long (e.g. 19646550).

6) Now ask them to think of any single digit in their number. Tell them to read out the whole number but when they get to the digit they are thinking of they should skip it (e.g. they might think of the number 4 and say 1966550)

7) You are now able to tell them what number they skipped.

How do you do this? When they read out the digits you should secretly add them all up in your mind (e.g. 1+9+6+6+5+5+0 = 32). When you have the total you should subtract the total from the next highest multiple of 9 (e.g. 36 – 32 = 4) – the answer will be the number they are thinking of.

## Why does this work?

The first 3 steps will always result in a multiple of 9. The difference between any number and the sum of all of its digits will always be a multiple of 9. When you multiply by any other number it will still result in a multiple of 9. Multiples of 9 have several interesting properties. The one we are interested in is the fact that if you add up the digits of any multiple of 9 the answer will always result in another multiple of 9 – check this, multiply 9 by any other digit and then add up all of the digits in your answer, now divide your new answer by 9 and you’ll see that it is divisible by 9. If you take a multiple of 9 and add the digits together and your new answer has more than one digit then add the digits together again and keep repeating until you have a single digit – it will always be a 9. When your friend misses out a digit the total will no longer add up to a multiple of 9 but the missing digit will be whatever is required to make this up to a multiple of 9.

## Exceptions

There are 2 cases where this won’t work. If they skip either a 0 or a 9 then the result will add up to a multiple of 9, so now you won’t know if they thought of a 0 or a 9. You have a couple of options:

1) Ask them to think of any digit except for 0. Now you know that if you get a multiple of 9 they must be thinking of 9.

2) When you get to the end and it is a multiple of 9 you could say “I’m getting the sense you are thinking of a very round number. It’s not a zero is it?” If they say yes you reply “I thought so” but if they say no then you can say “I didn’t think so, I’m getting the impression its round, but kind of curly, like a 6 or a 9. Focus on the number.... It’s a 9!”

## Other Methods

The most important part of the method is to get to a multiple of 9. There are many ways to do this. Here are a few more options, pick the one which appeals to you:

1) Ask them to write down any 4 digits, to scramble the order and subtract the smaller number from the larger number. Now they can multiply this number by any single digit and repeat until they get to a number with 7 or 8 digits.

2) Ask them to write down any 8 digits, to reverse the order and subtract the smaller number from the larger number. This will result in a multiple of 9.

3) Ask them to write down any 8 digits, now they can scramble these 8 digits into any order they like. Then subtract the smaller number from the larger number.

4) Interestingly this also works with phone numbers. Ask them to write down their phone number but to exclude the 0 from the beginning and then to scramble the digits into any order. Now they subtract the smaller number from the larger number.

## A Performance Tip

In any trick where you seem to read your friends mind you have to work out a way to make it interesting for other people who are watching. If your friend thinks of the number 1089 and you simply say “you are thinking of 1089!” then your friend will be the only person who knows you are correct. If you want to make this more interesting for anyone watching you need a way for them to know you have succeeded without having to rely on your friend to tell them. There are two main options for this:

1) The audience should all know the thought before you reveal it – you could ask your friend to whisper the number to everyone else, or to write it down and show it to everyone while your back is turned. The problem with this method is the audience might think you somehow heard what they said or secretly looked at the writing.

2) When you are “reading their mind” you say that you will write down what you are getting. You then write down the number, or word (you could potentially even draw it if it is something like a tractor) on a piece of paper but don’t let anyone see it. Then you can ask your friend to tell everyone what he is thinking of and then when you show everyone what you wrote the whole audience will be able to appreciate the trick at the same time! Once you’ve written the answer down I recommend handing it to your friend to hold (but make sure they don’t peek) – this is to ensure that you couldn’t somehow change what you wrote after they tell you what they were thinking of. Look back over the previous editions of Mathemagic, see if you can use this tip in any of the other tricks you’ve learnt.

## Puzzle

Finally I’d like to end with a puzzle, as this is the last edition of Mathemagic I will not be posting the solution. See if you can work it out.

Take 6 coins and arrange them into a cross with 4 in the vertical row and 3 in the horizontal row as in the picture below. Can you move just 1 coin to create a cross which has 4 coins in both rows?

Have fun performing your magic!