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

The ten letters, A-J, in the following two sections each contains a calculation. Which is the only letter that has the same answer in both sections?

- Section 1

E:6÷6 I:5+1 G:3+2 A:10–2 J:4+3 C:5×1 F:6÷3 B:8–4 H:9÷3 D:4×2

- Section 2

H:4–3 A:6÷1 I:3×3 B:6–1 F:6–4 J:8÷2 E:5×2 G:4×1 D:9–3 C:2+1

**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 3rd & 4th rows of the playing board contain the following fourteen numbers:

3 10 13 25 32 35 36 42 44 45 54 60 66 80

What is the difference between the two largest odd 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 NINE different ways to make **220 **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?

1 4 9 16 25 36 49 64 81 100

#*SquareNumbers*

**The Target Challenge**

Can you arrive at **220** by inserting **5**, **10**, **20** and **30** into the gaps on each line?

- (◯+◯)×◯+◯ = 220
- (◯+◯)×◯–◯ = 220
- ◯×◯+◯–double◯ = 220
- (◯+◯)²–half(◯–◯) = 220
- ◯×(◯÷◯)³–◯ = 220

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

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