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

Here’s a challenging number trail question involving seven arithmetical steps. Start with the number 139, then:

–8 +103 decrease by 50% –37 +109 1/3 of this ÷7 = **?**

What is your final answer?

**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 1st & 4th rows of the playing board contain the following fourteen numbers:

2 3 9 10 14 15 22 32 35 40 44 54 60 72

How many pairs of numbers have a sum of 75?

**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 TEN different ways to make **249 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **5**, **8** and **11 **once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

3 6 9 12 15 18 21 24 27 30

#*3TimesTable*

**The Target**** Challenge**

Can you arrive at **249** by inserting **1**, **2**, **3**, **4** and **5** into the gaps below?

- (◯×◯
^{◯}÷◯)–√◯ = 249

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

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