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

Carry out these instructions in our 7-step challenge:

- Work out the sum of 4 and 6,
- Calculate the product of 4 and 6,
- Find the difference between the two answers in Points 1 and 2,
- Square the answer to Point 3,
- Subtract 75 from your answer in Point 4,
- Find the square root of your answer in Point 5,
- Multiply your answer in Point 6 by 10.

What is your final answer?

**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 & 2nd rows contain the following fourteen numbers:

2 8 9 14 15 17 22 28 40 48 55 63 64 72

What is the product of the smallest and largest numbers on this list?

**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 SIX ways of making **110 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **1**, **4** and **9 **once each, with + – × ÷ available, which TWO numbers is it possible to make from the list below?

8 16 24 32 40 48 56 64 72 80

#*8TimesTable*

**The Target Challenge**

Can you arrive at **110** by inserting **6**, **7**, **8** and **10** into the gaps on both lines?

- (◯+◯)×◯+◯ = 110
- (◯+◯)×◯+double◯ = 110

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

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