**Th****e Main Challenge**

Using the three numbers **4**, **4** and **4** once each, with + – × ÷ available, there are just SIX target numbers from **1-30** that are mathematically possible to achieve. Can you find them?

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

4 11 13 24 25 27 30 36 42 45 66 70 77 80

Which number above 20, when 15 is subtracted from it, becomes a multiple of 7?

**The Lagrange Challenge**

*Lagrange’s Four-Square Theorem* states that every integer can be made by adding up to four square numbers.

For example, **7** can be made by **2²+1²+1²+1²** (4+1+1+1).

Show how you can make **153**, in TEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using the three digits **3**, **5** and **8** once each, with + – × ÷ available, which are the only TWO numbers it’s possible to make from the list below?

4 8 12 16 20 24 28 32 36 40

#*4TimesTable*

**The Target Challenge**

Can you arrive at **153** by inserting **3**, **4**, **6** and **7** into the gaps in each line below?

- (◯×◯–◯)×◯² = 153
- ◯+◯×treble(◯+◯) = 153

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

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