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

From the following list of eighteen numbers, eliminate all square numbers, multiples of 8, factors of 60 and prime numbers.

3 4 7 10 11 15 16 17 24 27 30 32 36 48 49 54 56 64

What is the sum of the TWO numbers that remain?

**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 & 7th rows contain the following fourteen numbers:

4 11 13 24 25 27 30 36 42 45 66 70 77 80

How many even numbers, when halved, become odd numbers?

**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** (4+1+1+1).

Show how you can make **154**, in TEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Based on our best-selling arithmetic board game.

Using the three digits **3**, **5** and **8** once each, with + – × ÷ available, which are the only TWO numbers it’s possible to make from the list below?

5 10 15 20 25 30 35 40 45 50

#*5TimesTable*

**The Target Challenge**

Can you arrive at **154** by inserting **5**, **6**, **7** and **8** into the gaps in each line below?

- (◯×◯–◯)×◯ = 154
- ◯²+◯²+◯²+◯ = 154

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

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