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

Using the numbers **5**, **5** and **10** once each, with + – × ÷ available, list the NINE target numbers from **1-30** that are mathematically possible to achieve.

**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 & 7th rows of the playing board contain the following fourteen numbers:

4 6 7 11 16 21 24 27 30 50 70 77 81 84

Which four different numbers have a sum of 100?

**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 ELEVEN different ways to make **245 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **2**, **5** and **10 **once each, with + – × ÷ available, which are the only THREE numbers it is possible to make from the list below?

15 30 45 60 75 90 105 120 135 150

#*15TimesTable*

**The Target**** Challenge**

Can you arrive at **245** by inserting **1**, **2**, **3**, **4** and **5** into the gaps below?

- (◯+◯)²×◯×(◯–◯) = 245

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

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