**T**** h****e Main Challenge**

If you multiply a mystery number by 6 and then subtract 6, the result is the same as if you first multiplied the same mystery number by 3 and then added 3.

Is the value of this mystery number **1**, **2**, **3**, **4** or **5**?

**The 7puzzle Challenge**

The playing board of **the 7puzzle game** is a 7-by-7 grid containing 49 different numbers, ranging from **2 **up to **84**.

The 3rd & 4th rows contain the following fourteen numbers:

3 10 13 25 32 35 36 42 44 45 54 60 66 80

What is the sum of the multiples of 7 listed?

**The Lagrange Challenge**

*Lagrange’s Four-Square Theorem* states that every positive integer can be made by adding up to four square numbers.

For example, **7** can be made by **2²+1²+1²+1²** (or 4+1+1+1).

There are just THREE ways of making **112 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **1**, **4** and **9 **once each, with + – × ÷ available, which TWO numbers is it possible to make from the list below?

1 4 9 16 25 36 49 64 81 100

#*SquareNumbers*

**The Target Challenge**

Can you arrive at** 112** by inserting **3**, **5**, **10** and **11** into the gaps on both lines?

- ◯×◯+◯–◯ = 112
- (◯+◯)×(◯–◯) = 112

**An****swers **can be found **here**.

**Click Paul Godding for details of online maths tuition.**