### Plum Tree

Label this plum tree graph to make it totally magic!

### Magic W

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

### 2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

# Back to Basics

##### Stage: 4 Challenge Level:

In March we posed the problem:
"The number $3723$ (in base $10$) is written as $123$ in another base.
What is that base?" ............ The answer to this can be found in the March problem archive.

We could have written this question as:

Find b where $3723_{10} = 123_{b}$

So, moving on ...................

$123_{20}$ is $1 \times 20^2 + 2 \times 20 + 3 = 443_{10}$

$123_{21}$ is $1 \times 21^2 + 2 \times 21 + 3 = 486_{10}$

$123_{22}$ is $1 \times 22^2 + 2 \times 22 + 3 = 531_{10}$

$531 - 486 = 45$

$486 - 443 = 43$

Investigate these differences when $123_{b}$ is converted to base $10$ (for different values of $b$).

Try to explain what is happening.