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

Group these ten numbers into **five pairs** so that the difference between the two numbers in each pair is divisible by **7**:

6 17 28 37 45 58 64 78 83 98

**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 & 6th rows contain the following fourteen numbers:

5 12 13 18 20 25 33 36 42 45 49 56 66 80

From the list, what is the sum of the multiples of 7?

**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 SEVEN ways of making **133 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **3**, **6** and **10 **once each, with + – × ÷ available, which THREE numbers is it 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 **133** by inserting **10**, **11**, **13** and **20** into the gaps on each line?

- ◯×◯+◯–◯ = 133
- ◯×◯+√(◯–◯) = 133

**Ans****wers **can be found **here**.

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