Figures of fun: Amaze your friends with these fantastic maths magic tricks

• By Zaheer Khan

Peter M Higgins, author and numbers man, reveals the magic behind something many of us struggle to get our heads around

A maths wizard has discovered the largest known prime number and it's an amazing 17 MILLION digits long.
Curtis Cooper, of the University of Central Missouri, in the US found it by using hundreds of computers networked together.
You might not have to be a maths genius to know that a prime is a number greater than 1, which can only be divided by itself and 1, like 2, 3, 5, 7, 11, but to find one that big, GCSE maths would be a good start.
However, in the numbers world, size doesn't matter when it comes to the fascinating and bizarre.
Here Peter M Higgins, author and numbers man, reveals the magic behind something many of us struggle to get our heads around...

Hailstone numbers

There are many simple questions about numbers that no one has been able to answer.
Start with any number - if it is even divide it by 2, if odd multiply by 3 and add 1 then keep going, writing down the sequence of numbers that you generate. For example, starting with 7 we are led by these rules through the sequence:
7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.
It seems no matter what number you start with you eventually hit a 1. These sequences are called the "hailstone numbers" because, like hailstones, they go up and down a number of times before inevitably falling to Earth. However, no one has been able to prove that this has to happen every time.

Twos, threes & fives

Think of a number. Add 4, then multiply the result by 4. Subtract 8, then divide the result by 4. Finally take away your original secret number. The answer is 2.
Think of another number.
Double it. Add 9. Subtract 3. Divide by 2. Subtract your original number. The answer is 3.
Think of any three-digit number.
Add 7. Multiply by 2.
Subtract 4, then divide the result by 2.
Subtract it from the original number you thought of.
The answer is 5.

Ninety nine

Write down any two different numbers from 1 to 9. Then reverse the two numbers.
You should have two two-digit numbers.
Subtract the smaller number from the larger one.
Take the result, reverse the digits, and add that number to the one you got when you subtracted.
The answer is 99.
For example: 72 reverses to make 27.
Subtract the smaller (27) from the larger (72): 45.
Reverse these digits to make 54.
Add this to the previous number.
The answer is 99.

Threesy does it

You can discover whether a number is a multiple of 3 just by checking whether this is true for the sum of its digits.
For example, 12,894 has 1 + 2 + 8 + 9 + 4 = 24 = 3 x 8, so 12,894 is a multiple of 3.
You don't need to do the long division in order to find this out.
You can do this even for huge numbers that your calculator could never cope with.
For example, try: 111,222,333,444,555,666,777, 888,987. Is it divisible by 3? In fact, if you're clever, you might be able to give the answer before summing the digits.

10% up then 10% down means you lose out

A worker's boss explains that in order to stay competitive he will have to cut his pay by 10% but he will allow the employee to work 10% more hours to make up for it, "so your pay will be maintained".
Afraid not! If the worker was being paid, say, £100, the 10% cut takes him down to £90. The 10% extra hours will add back on 10% of £90, which gives him £99. He is still £1 worse off. Beware percentages - you need to know what they refer to.

Never-ending squares

Square numbers (the products of numbers multiplied by themselves) and prime numbers are important and your internet security only works because the prime numbers never run out.
You can get the endless list of squares just by adding the odd numbers up: 1 = 1 x 1, 1 + 3 = 4 = 2 x 2, 1 + 3 + 5 = 9 = 3 x 3, 1 + 3 + 5 + 7 = 16 = 4 x 4 ... and this pattern never lets you down.
However, when it comes to primes, we still have to go out hunting for them, which is why at any one time there is always a world champion largest known prime.

'Mind reading' trick

Choose a single-digit number, multiply it by 9 and if the answer has two digits add them together.
Subtract 5 from what you have, giving you a number. Turn the number into a letter by the rule A = 1, B = 2 and so on. Think of a country beginning with your letter. Take the last letter of your country and think of an animal that begins with that letter. It's odds on that you have a kangaroo in Denmark.

It all adds up... to 9

1x9=09 =0+9=9
2x9=18 =1+8=9
3x9=27 =2+7=9
4x9=36 =3+6=9
5x9=45 =4+5=9
6x9=54 =5+4=9
7x9=63 =6+3=9
8x9=72 =7+2=9
10 x 9 = 90 = 9 + 0 = 9

Magic cube

  Think you can conjure the right numbers? Try this little trick then.
Label the corners of the cube on the left with different numbers in the range 1 to 8 so that each face adds up to the same total.
The solution is on the right.

One, two, three

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10 = 1111111111
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