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

When playing **Mathematically Possible**, players must analyse which numbers can (or can’t) be made from the three numbers rolled on their dice.

Using the numbers **3**, **4** and **6**, with + – × ÷ available, which THREE of the following target numbers are NOT mathematically possible to achieve?

1 2 3 5 6 7 8 10 12 13 14 18 21 22 24 27 30

**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 4th & 5th rows of the playing board contain the following fourteen numbers:

3 6 7 10 16 21 32 35 44 50 54 60 81 84

What is the difference between the sum of the multiples of 5 and sum of the multiples of 6?

**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).

Show how you can make **193**, in EIGHT different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **3**, **4** and **12** once each, with + – × ÷ available, which SIX numbers is it possible to make from the list below?

12 24 36 48 60 72 84 96 108 120

#*12TimesTabl**e*

**The Target ****Challeng****e**

Can you arrive at **193** by inserting **7**, **10**, **11** and **13** into the gaps on each line?

- ◯×(◯+◯)+◯ = 193
- ◯²+◯²+◯+◯ = 193

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

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