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

Group the following numbers into three lots of three so the products of each of the triples are the same. What is this product?

3 4 5 6 7 8 28 30 35

**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 5th & 6th rows of the playing board contain the following fourteen numbers:

5 6 7 12 16 18 20 21 33 49 50 56 81 84

From the list, which pair of numbers have a sum of 77?

**T****he 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 NINE different ways to make **221 **when using *Lagrange’s Theorem*. How many can you find?

**The Mathematically Possible Challenge**

Using **4**, **6** and **9 **once each, with + – × ÷ available, which are the only THREE numbers it is 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 **221** by inserting **5**, **6**, **7** and **10** into the gaps on each line?

- (◯+◯)²–◯×◯ = 221
- (◯+◯+1)×(◯+◯+1) = 221

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

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