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

Using the three numbers **1**, **2** and **4** just once each, with + – × ÷ available to you, what is the lowest positive number it is NOT possible to make?

**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 TWO numbers listed have exactly four factors each?

**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 **152**, in THREE different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using the three digits **2**, **6** and **9** once each, with + – × ÷ available, which are the only FOUR numbers it’s possible to make from the list below?

2 3 5 7 11 13 17 19 23 29

#*PrimeNumbers*

**The Target Challenge**

Can you arrive at **152** by inserting **4**, **8**, **10** and **12** into the gaps on each line?

- ◯×◯+◯×◯ = 152
- (◯+◯)×◯–◯ = 152

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

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