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

You are playing our **Mathematically Possible **board game and have rolled the numbers **6**, **6** and **6** with your three dice. Using these once each, with + – × ÷ available, which SIX target numbers from **1-30** is it possible to make?

**T****he**** 7puzzle Challenge**

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

The 3rd & 5th rows contain the following fourteen numbers:

6 7 13 16 21 25 36 42 45 50 66 80 81 84

What is the difference between the sum of the multiples of 7 and the sum of the multiples of 9?

**The Lagrange Challenge**

*Lagrange’s Four-Square Theorem* states that every 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).

Show how you can make **172**, in EIGHT different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **4**, **6** and **10** once each, with + – × ÷ available, which are the SIX target numbers it’s possible to make from the list below?

1 4 9 16 25 36 49 64 81 100

#*SquareNumber*s

**The Target Challenge**

Can you arrive at **172** by inserting **3**, **4**, **5** and **8** into the gaps on each line?

- (◯+◯×◯)×◯ = 172
- (◯+◯)²+◯²+◯ = 172

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

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