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

Study the seven clues and place the numbers **1-9** into the nine positions on this 3-by-3 grid. Each number should appear exactly once:

**x x x**

**x x x**

**x x x**

Clues:

- The 8 is directly above the 5,
- The 6 is further right than the 7,
- The 7 is further right than the 1,
- The 1 is lower than the 5,
- The 5 is further right than the 9,
- The 3 is higher than the 9 and further right than the 2,
- The 4 is higher than the 7 and further right than the 8.

**The**** 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 2nd & 7th rows contain the following fourteen numbers:

4 8 11 17 24 27 28 30 48 55 63 64 70 77

What is the difference between the sum of the highest and lowest even numbers and the product of the highest and lowest even numbers?

**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** (or 4+1+1+1).

Show how you can make **170**, in ELEVEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **4**, **6** and **10** once each, with + – × ÷ available, which is the ONLY target number it’s possible to make from the list below?

9 18 27 36 45 54 63 72 81 90

#*9TimesTable*

**The Target Challenge**

Can you arrive at **170** by inserting **9**, **10**, **20** and **20** into the gaps on each line?

- ◯+◯×◯–◯ = 170
- (◯–◯÷◯)×◯ = 170

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

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