**T**** he Main**** Challenge**

Of the numbers **1**,** 2**,** 3 … 6000**, how many are NOT multiples of 2, 3 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 4th & 5th rows contain the following fourteen numbers:

3 6 7 10 16 21 32 35 44 50 54 60 81 84

What is the average of the two prime numbers on the list?

**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 FIVE ways of making **89 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **2**, **3** and **11 **once each, with + – × ÷ available, which THREE numbers is it possible to make from the list below?

30 31 32 33 34 35 36 37 38 39

#*NumbersIn30s*

**The Target Challenge**

Can you arrive at **89** by inserting **1**, **4**, **8** and **10** into the gaps on each line?

- ◯²+◯²+◯–◯ = 89
- ◯×◯+(◯–◯)² = 89
- (◯+◯)³–double(◯+◯) = 89

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

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