### Exploring Wild & Wonderful Number Patterns

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

### I'm Eight

Find a great variety of ways of asking questions which make 8.

### Dice and Spinner Numbers

If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

# Abundant Numbers

##### Stage: 2 Challenge Level:

Georgia, from Holy Trinity School, found two more abundant numbers:

$36$: total is $55$ ($1+2+3+4+6+9+12+18$)
$70$: total is $74$ ($1+2+5+7+10+14+35$)

Rachael, Jamie, Heledd, Sian, Dafydd, Tom, Edward and Isaac from Ysgol Bryncrug clearly worked hard on this problem.  They told us:

We decided to find out which of the numbers from $1$ to $100$ are abundant numbers.
We decided that prime numbers are not abundant numbers:
$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$
because without themselves $1$ is their only factor.

We crossed out all the prime numbers from our $100$ square:

Then we looked at all the remaining numbers. Here is an example:

$27 = 1\times27$, $3\times9$
So the factors of $27$ are:
$1, 3, 9, 27$
The sum of its factors without itself is:
$1+3+9 = 13$

$13$ is less than $27$ therefore $27$ is not an abundant number

And another example:
$48 = 1\times48$, $2\times24$, $3\times16$, $4\times12$, $6\times8$
So the factors of $48$ are:
$1, 2, 3, 4, 6, 8, 12, 16, 24, 48$.
The sum of its factors without itself is:
$1+2+3+4+6+8+12+16+24 = 76$

$76$ is greater than $48$ therefore $48$ is an abundant number

Something different happened with $6$:
$6 = 1\times6$, $2\times3$
So the factors of $6$ are:
$1, 2, 3$ and $6$
The sum of its factors without itself is: $1+2+3 = 6$

$6$ is equal to $6$!!

This also happened with $28$:
$1+2+4+7+14 = 28$

Edward wanted to find out what we call a number where the sum of its factors (without itself) is equal to the number. He asked his Dad and he told him that it is called a PERFECT NUMBER.
He also found out that the next perfect number after $28$ is $496$. Then $8128$!!

We continued marking the numbers on the $100$ square:

There are twenty two abundant numbers on our $100$ square.

Thank you for letting us know how you approached this problem.  I like the way you discovered perfect numbers along the way.  Well done!