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

We invite you to use seven 3’s (**3 3 3 3 3 3** and **3**) once each, with + – × ÷ available, to make various target numbers.

For instance, to make **1** and **2**, you could do:

- 3 × (3÷3) – (3÷3) – (3÷3) =
**1** - (3+3) × (3÷3) × (3÷3) ÷ 3 =
**2**. . . and so on.

Continuing, as above, can you make the target numbers from **3** to **6**?

**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 of the playing board contain the following fourteen numbers:

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

Find three different numbers from the list that 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 THIRTEEN different ways to make **243 **when using *Lagrange’s Theorem*. How many of them can you find?

**The Mathematically Possible Challenge**

Using **2**, **5** and **10 **once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

12 24 36 48 60 72 84 96 108 120

#*12TimesTable*

**The Target**** Challenge**

Can you arrive at **243** by inserting **2**, **3**, **4**, **5** and **6** into the gaps below?

- ◯×(◯+◯)²+(◯+◯)² = 243

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

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