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

As an example, both **4** and **8** can be written as the sum of two prime numbers, as shown here:

- 4 = 2+2
- 8 = 5+3

… but how many numbers **less than 20** CANNOT be written as the sum of two prime numbers?

**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 & 7th rows of the playing board contain the following fourteen numbers:

2 4 9 11 14 15 22 24 27 30 40 70 72 77

Which four different numbers above have a sum of 100?

**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 TEN different ways to make **228 **when using *Lagrange’s Theorem*. How many can you find?

**The Mathematically Possible Challenge**

Using **2**, **4** and **8 **once each, with + – × ÷ available, which FOUR numbers is it 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 **228** by inserting **3**, **4**,** 5** and **6** into the gaps on each line?

- ◯×(◯²+◯²–◯) = 228
- ◯×(◯²×◯+double◯) = 228

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

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