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

From the numbers in the range **1-50**, eliminate the following:

- multiples of 3
- square numbers
- prime numbers
- even numbers

Which number is the only one that remains?

**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

Which number, when 20 is added to it, becomes a square number?

**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 SEVEN different ways to make **199 **when using *Lagrange’s Theorem*. How many can you find?

**The Mathematically Possible Challenge**

Using **5**, **7** and **11 **once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

4 8 12 16 20 24 28 32 36 40

#*4TimesTable*

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

Can you arrive at **199** by inserting **11**, **12**, **14** and **20** into the gaps on each line?

- ◯×◯+◯+◯ = 199
- ◯²+◯+◯–◯ = 199

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

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