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

There is just one set of three consecutive numbers in ascending order whose sum is less than 50 and follow this sequence:

- prime number – cube number – square number

What is the sum of these three numbers?

**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 6th & 7th rows of the playing board contain the following fourteen numbers:

4 5 11 12 18 20 24 27 30 33 49 56 70 77

What is the sum of the multiples of 11?

**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).

Show how you can make **197**, in SEVEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **3**, **4** and **12** once each, with + – × ÷ available, which FOUR numbers is it possible to make from the list below?

2 3 5 7 11 13 17 19 23 29

#*PrimeNumbers*

**The Target ****Challeng****e**

Can you arrive at **197** by inserting **1**, **4**, **11** and **14** into the gaps on each line?

- ◯×(◯+◯)–◯ = 197
- (◯+◯)²–(◯×double◯) = 197

**An****swers **can be found **here**.

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