**The 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**

Based on our best-selling arithmetic board game.

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

**Answers **can be found **here**.

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