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

This is the 3rd and final part of a number puzzle posted initially on **DAY 10**, then on **DAY 180**, and made famous by French writer, George Perec.

The challenge involves using seven 7′s (7, 7, 7, 7, 7, 7 and 7) once each, with + – × and ÷ available, to make various target numbers.

For instance, to make **9** in the previous challenge on **DAY 180**, you could have done:

- [7 + (7÷7) + (7÷7)] × (7÷7) = 9

Up until now, it has been possible to make every target number from 1 through to 9 with seven 7’s, so for today’s time-consuming task:

**Part 1**: Show how to make all target numbers from 10 through to 19.

**Part 2**: Which is the first number after 19 that is impossible to make with seven 7’s?

**The**** 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 2nd & 3rd rows contain the following fourteen numbers:

8 13 17 25 28 36 42 45 48 55 63 64 66 80

Which two numbers have a difference of 13?

**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 **190**, in EIGHT different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

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

8 16 24 32 40 48 56 64 72 80

#*8TimesTabl**e*

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

Can you arrive at **190** by inserting **4**, **5**, **6** and **14** into the gaps on each line?

- (◯+◯)×(◯+◯) = 190
- (◯×◯+◯)×◯ = 190
- ◯²–◯×(◯–◯) = 190

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

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