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

Which is the lowest whole number that is NOT a multiple of **4**, **5** or **6**, nor a prime number, square number or cube number?

**The 7puzzle Challenge**

The playing board of **the 7puzzle game** is a 7-by-7 grid containing 49 different numbers, ranging from **2 **up to **84**.

The 2nd & 4th rows contain the following fourteen numbers:

3 8 10 17 28 32 35 44 48 54 55 60 63 64

Which odd number, when 1 is subtracted from it, becomes a prime number?

**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 TWO ways of making **21 **when using *Lagrange’s Theorem*. Can you find both?

**The Mathematically Possible Challenge**

Using **5**, **6** and **8 **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 **21** by inserting **3**, **4**, **5** and **6** into the gaps on each line?

- (◯+◯–◯)×◯ = 21
- ◯×◯+√(◯+◯) = 21
- ◯²–(◯×◯÷◯)² = 21

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

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