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

Using the numbers **4**, **5** and **6** once each, together with + – × ÷, which THREE target numbers from the following list are NOT mathematically possible to make?

2 3 4 5 6 7 10 12 14 15 19 20 26 29 30

**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

What is the difference between the highest and lowest multiples of 7?

**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 FIVE ways of making **50 **when using *Lagrange’s Theorem*. Can you find them all?

**The Mathematically Possible Challenge**

Using **8**, **9** and **10 **once each, with + – × ÷ available, which is the ONLY number it is 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 **50** by inserting **2**, **5**, **10** and **20** into the gaps on each line?

- ◯²–◯×◯÷◯ = 50
- (◯+◯)×◯+◯ = 50
- ◯⁵+◯–◯÷◯ = 50

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

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