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

It is possible to use seven 4’s (**4 4 4 4 4 4** and **4**) once each, together with the four operations + – × ÷, to make all the different target numbers **1**, **2**, **3** and **4**.

For instance, one way of arriving at the number **1** is:

4 – 4÷4 – 4÷4 – 4÷4 = **1**

Can you show how to make the next three target numbers **2**, **3** and **4**?

**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 4th & 7th rows contain the following fourteen numbers:

3 4 10 11 24 27 30 32 35 44 54 60 70 77

What is the sum of the multiples of 9?

**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 **76 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **4**, **6** and **12 **once each, with + – × ÷ available, which TWO numbers is it possible to make from the list below?

70 71 72 73 74 75 76 77 78 79

#*NumbersIn70s*

**The Target Challenge**

Can you arrive at **76** by inserting **6**, **8**, **10** and **12** into the gaps on each line?

- ◯×◯+(◯–◯)² = 76 (2 different ways)
- ◯×◯–(◯÷◯)² = 76
- ◯×(◯–◯)+◯² = 76

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

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