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

From the numbers **1-20**, eliminate all:

- square numbers
- prime numbers
- triangular numbers
- multiples of 6

Add together the numbers that remain; what is your total?

**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 1st & 4th rows contain the following fourteen numbers:

2 3 9 10 14 15 22 32 35 40 44 54 60 72

From this list, what is the sum of the multiples of 8?

**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 FIVE ways of making **142 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using the three digits **2**, **6** and **9** once each, with + – × ÷ available, which are the only TWO numbers it’s possible to make from the list below?

5 10 15 20 25 30 35 40 45 50

#*5TimesTable*

**The Target Challenge**

Can you arrive at **142** by inserting **4**, **9**, **10** and **13** into the gaps on each line?

- ◯×◯+◯×◯ = 142
- ◯×◯+◯×√◯ = 142

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

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