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

You have the same starting number and final answer, **both 22**.

There are 10 arithmetical steps altogether but the 10th, and final, step is missing. If this final step involves a whole number, what should it be to make the final answer **22**?

+2 ÷6 ×4 –3 ×2 +4 ÷5 +6 ×2 **?** = **22**

**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 3rd & 5th rows contain the following fourteen numbers:

6 7 13 16 21 25 36 42 45 50 66 80 81 84

List three sets of three numbers that all have a sum of 100, the numbers in each set must be different.

**The 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 **171**, in TEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **4**, **6** and **10** once each, with + – × ÷ available, which are the only TWO target numbers it is possible to make from the list below?

10 20 30 40 50 60 70 80 90 100

#*10TimesTable*

**The Target Challenge**

Can you arrive at **171** by inserting **3**, **6**, **9** and **10** into the gaps on each line?

- ◯×◯×◯–◯ = 171
- (◯+◯+◯)×◯ = 171

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

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