**T****he Main Challenge**

The smallest positive integer that has exactly three factors is **FOUR**; these are 1, 2 and 4. Find the next integer to have just three factors and the product of these three numbers.

**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 3rd & 7th rows contain the following fourteen numbers:

4 11 13 24 25 27 30 36 42 45 66 70 77 80

What is the sum of the multiples of 10?

**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 **46 **when using *Lagrange’s Theorem*. Can you find both?

**The Mathematically Possible Challenge**

Using **8**, **9** and **10 **once each, with + – × ÷ available, which is the ONLY number it is possible to make from the list below?

7 14 21 28 35 42 49 56 63 70

#*7TimesTables*

**The Target Challenge**

Can you arrive at **46** by inserting **3**, **4**, **5** and **9** into the gaps on each line?

- (◯+◯)×◯+◯ = 46
- ◯²+◯×◯+◯ = 46
- ◯×◯+◯–√◯ = 46

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

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