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

It is possible to use seven 5’s (**5 5 5 5 5 5** and **5**) once each, with the four operations + – × ÷, to make all the target numbers from **1 to 5**.

For instance, to arrive at the target numbers **1** and **2**, you can do:

- [(5+5)÷5 – 5÷5] × 5÷5 =
**1** - [(5+5)÷5 – 5÷5] + 5÷5 =
**2**

Your task is to show how to arrive at the other target numbers **3**, **4** and **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 5th & 7th rows contain the following fourteen numbers:

4 6 7 11 16 21 24 27 30 50 70 77 81 84

What is the product of the two prime numbers listed?

**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 SIX ways of making **137 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **3**, **6** and **10 **once each, with + – × ÷ available, which TWO numbers is it possible to make from the list below?

1 4 9 16 25 36 49 64 81 100

#*SquareNumbers*

**The Target Challenge**

Can you arrive at **137** by inserting **4**, **5**, **6** and **8** into the gaps on each line?

- ◯³+◯²+◯×◯ = 137
- ◯³+◯×◯÷◯ = 137

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

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