**The Main 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** (4+1+1+1).

Show how you can make **31** when using *Lagrange’s Theorem* in TWO different ways.

**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 1st & 4th rows of the playing board contain the following fourteen numbers:

2 3 9 10 14 15 22 32 35 40 44 54 60 72

What is the sum of the multiples of 5?

**The Mathematically Possible Challenge**

Based on our best-selling arithmetic board game.

Using the three digits **2**, **6** and **9** once each, with + – × ÷ available, which are the only THREE numbers it’s possible to make from the list below?

8 16 24 32 40 48 56 64 72 80

#*8TimesTable*

**The Target Challenge**

Can you arrive at **145** by inserting **3**, **4**, **5** and **8** into the gaps on each line?

- (◯×◯–◯)×◯ = 145
- (◯+◯)²–double(◯×◯) = 145
- (◯+◯)²+half(◯–◯) = 145

**Answers **can be found **here**.

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