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

From the numbers below, eliminate all:

- square numbers,
- triangular numbers,
- multiples of 4 and
- factors of 70.

1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 25

Which is the only number left remaining?

**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 1st & 5th rows contain the following fourteen numbers:

2 6 7 9 14 15 16 21 22 40 50 72 81 84

What is the difference between the lowest and highest multiples of 5?

**The 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 THREE ways of making **53 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **8**, **9** and **10 **once each, with + – × ÷ available, which is the ONLY number it is possible to make from the list below?

70 71 72 73 74 75 76 77 78 79

#*NumbersIn70s*

**The Target Challenge**

Can you arrive at **53** by inserting **4**, **4**, **5** and **6** into the gaps on each line?

- (◯+◯)×◯+◯ = 53
- ◯²+◯×◯+◯ = 53
- ◯²×◯–(◯+◯) = 53

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

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