**Th**** e Main Challenge**

Pair the ten numbers below so that the difference between the two numbers in each pair is exactly 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 1st & 3rd rows of the playing board contain the following fourteen numbers:

2 9 13 14 15 22 25 36 40 42 45 66 72 80

What is the sum of the factors of 80 listed above?

**T****he 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 NINE different ways to make **205 **when using *Lagrange’s Theorem*. How many can you find?

**The Mathematically Possible Challenge**

Using **5**, **7** and **11 **once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

10 20 30 40 50 60 70 80 90 100

#*10TimesTable*

**The Target ****Challeng****e**

Can you arrive at **205** by inserting **2**, **10**, **11 **and **15** into the gaps on each line?

- (◯×◯×◯)–◯ = 205
- (◯+◯÷◯)×◯ = 205

**A****nswers **can be found **here**.

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