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

This is a continuation of a number puzzle, posted on **DAY 10**, made famous by French writer, George Perec.

In that challenge, we asked you to use seven 7’s (7, 7, 7, 7, 7, 7 and 7) once each, with + – × and ÷ available, to make target numbers from **1 **to** 3**.

To continue the puzzle, and make **4**, you could simply do:

- 7 – 7÷7 – 7÷7 – 7÷7 = 4

We now invite you to make all the target numbers from **5** through to **9**.

**T****he**** 7puzzle Challenge**

The playing board of **the 7puzzle game** is a 7-by-7 grid of 49 different numbers, ranging from **2 **up to **84**.

The 1st & 6th rows contain the following fourteen numbers:

2 5 9 12 14 15 18 20 22 33 40 49 56 72

Which multiple of 8, when 7 is subtracted from it, becomes a square number?

**T****he Lagrange Challenge**

*Lagrange’s Four-Square Theorem* states that every 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 **180**, in TEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **3**, **6** and **12** once each, with + – × ÷ available, which is the ONLY target number it is 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 **180** by inserting **4**, **6**, **8** and **10** into the gaps on each line?

- (◯+◯+◯)×◯ = 180
- (◯+◯)×(◯+◯) = 180
- ◯²+◯×(◯+◯) = 180
- (◯+◯+double◯)×◯ = 180

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

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