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

The two sections below both contain ten letters, A-J; each of which has a multiplication calculation assigned to it:

- Section 1

C:10×2 J:4×3 F:8×5 D:3×3 I:3×2 G:7×4 B:5×4 H:9×4 E:10×3 A:6×4

- Section 2

B:12×2 D:6×1 G:8×3 J:5×2 E:6×5 I:6×6 H:8×6 A:9×2 F:7×2 C:6×3

Which is the only letter that contains the same answer in both sections?

**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 5th & 7th rows contain the following fourteen numbers:

4 6 7 11 16 21 24 27 30 50 70 77 81 84

Find three different numbers that have a sum of 77?

**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 TEN ways of making **138 **when using *Lagrange’s Theorem*. Can you find them all?

**The Mathematically Possible Challenge**

Using **3**, **6** and **10 **once each, with + – × ÷ available, which FOUR numbers is it 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 **138** by inserting **2**, **6**, **10** and **15** into the gaps on each line?

- ◯×◯–◯×◯ = 138
- (◯+◯–◯)×◯ = 138
- ◯²+double◯+◯+◯ = 138

**An****swers **can be found **here**.

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