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

If the number sequence **5 9 13 17 21 … ** is continued, which is the only number from the following list that appears later in the sequence?

35 43 59 67 71 85 95

**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 3rd & 5th rows contain the following fourteen numbers:

6 7 13 16 21 25 36 42 45 50 66 80 81 84

Is the sum of the even numbers more than double the sum of the odd numbers?

**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 FOUR ways of making **69 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **4**, **6** and **12 **once each, with + – × ÷ available, which TWO numbers are NOT possible to make from the list below?

6 12 18 24 30 36 42 48 54 60

#*6TimesTable*

**The Target Challenge**

Can you arrive at **69** by inserting **2**, **3**, **4** and **7** into the gaps on each line?

- ◯²×◯+◯+◯ = 69
- ◯³+(◯+◯)÷◯ = 69
- (◯²+◯)×◯–double◯ = 69

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

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