**T**** he**** Main Challenge**

A *Keith Number*, made famous by **Mike Keith**, is worked out in a not-too-dissimilar way to *Fibonacci Numbers*. If you like playing around with numbers, have a go at this fun concept. The first 2-digit *Keith Number*, **14**, is worked out as follows:

**Try 14**: 1+4=5; 4+5=9; 5+9=**14**(the total arrives back to the original number).

By following this pattern, can you find the next 2-digit *Keith Number*?

[Hint: it’s not too far away!]

**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 three numbers, when 6 is added to them, each become multiples of 7?

**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 **187**, in NINE different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **3**, **4** and **12** once each, with + – × ÷ available, which THREE numbers are not possible to make from the list below?

4 8 12 16 20 24 28 32 36 40

#*4TimesTabl**e*

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

Can you arrive at **187** by inserting **4**, **5**, **7** and **9** into the gaps on each line?

- (◯×◯×◯)+◯ = 187
- ◯²×◯+◯×√◯ = 187

**A****nswers **can be found **here**.

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