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

You have been given the task of manually numbering a 100-page document from **1 to 100**.

Which digit will appear most, and how many times?

**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 of the playing board contain the following fourteen numbers:

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

What is the difference between the highest and lowest square numbers?

**T****he 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 TEN different ways to make **213 **when using *Lagrange’s Theorem*. How many can you find?

**The Mathematically Possible Challenge**

Using **4**, **6** and **9 **once each, with + – × ÷ available, which is the ONLY number it is possible to make from the list below?

4 8 12 16 20 24 28 32 36 40

#*4TimesTable*

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

Can you arrive at **213** by inserting **3**, **4**, **5** and **15** into the gaps on each line?

- (◯×◯–◯)×◯ = 213
- ◯²–(◯+◯+◯) = 213
- (◯+double◯)×◯+◯ = 213

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

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