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

You have 12 balls and each ball is numbered differently from **1 to 12**. They are randomly placed into two bags so each bag contains the six numbered balls shown below:

- Red bag:
**2 3 4 6 9 11** - Blue bag:
**1 5 7 8 10 12**

You then move a ball from the red bag to the blue bag. The total of the seven balls in the blue bag is now **double** the total of the five balls left in the red bag.

Which ball did you move from the red bag to the blue bag?

**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 4th & 6th rows contain the following fourteen numbers:

3 5 10 12 18 20 32 33 35 44 49 54 56 60

How many pairs of numbers have a difference of 12?

**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 **164**, in SIX 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 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 **164** by inserting **4**, **6**, **7** and **8** into the gaps on each line?

- (◯×◯–◯)×◯ = 164
- ◯²×◯–quarter◯³–√◯ = 164

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

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