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

Can you insert the numbers **1-9**, exactly once each, into the gaps below so that all three lines work out arithmetically?

◯ + ◯ = ◯

◯ – ◯ = ◯

◯ ÷ ◯ = ◯

**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 & 3rd rows contain the following fourteen numbers:

8 13 17 25 28 36 42 45 48 55 63 64 66 80

Which number, when 20 is added to it, becomes a square number?

**T****he 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 **189**, in ELEVEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **3**, **4** and **12** once each, with + – × ÷ available, which SIX numbers is it possible to make from the list below?

6 12 18 24 30 36 42 48 54 60

#*6TimesTabl**e*

**The Target ****Challeng****e**

Can you arrive at **189** by inserting **3**, **4**, **5** and **7** into the gaps on each line?

- ◯×◯×(◯+◯) = 189
- (◯+◯–◯)³+◯³ = 189

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

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