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

Firstly, write down three separate lists only containing numbers in the range **1 to 100**:

- List 1 – multiples of 9
- List 2 – factors of 108
- List 3 – triangular numbers

**Part 1**: Which is the only number present on all three lists?

**Part 2**: List the eight other numbers that are on exactly two of the lists?

**Part 3**: How many DIFFERENT numbers are written on the three lists?

**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 2nd & 4th rows of the playing board contain the following fourteen numbers:

3 8 10 17 28 32 35 44 48 54 55 60 63 64

From the list, what is the sum of the even 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 EIGHT different ways to make **233 **when using *Lagrange’s Theorem*. How many of them can you find?

**The Mathematically Possible Challenge**

Using **2**, **4** and **8 **once each, with + – × ÷ available, which is the ONLY number it is possible to make from the list below?

2 3 5 7 11 13 17 19 23 29

#*PrimeNumbers*

**The Target Challenge**

Can you arrive at **233** by inserting **11**, **12**, **13**, **14** and **15** into the gaps below?

- 6×◯+5×◯+4×◯+3×◯–◯ = 233

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

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