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

Can you arrive at the target number **105** in two different ways when using the five numbers **1**, **2**, **3**, **4** and **5** once in each calculation, and with + – × ÷ available?

**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 & 5th rows contain the following fourteen numbers:

6 7 13 16 21 25 36 42 45 50 66 80 81 84

What is the difference between the highest multiple of 9 and highest multiple of 11?

**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 FOUR ways of making **68 **when using *Lagrange’s Theorem*. Can you find them all?

**The Mathematically Possible Challenge**

Using **4**, **6** and **12 **once each, with + – × ÷ available, which is the ONLY number it is 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 **68** by inserting **2**, **3**, **6** and **8** into the gaps on each line?

- (◯+◯)×◯+◯ = 68
- ◯²×(◯+◯+◯) = 68
- (◯+◯)×◯–◯² = 68

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

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