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

Read the facts below:

- I am a 2-digit number,
- I am a multiple of 7,
- The difference between my two digits is 4.

Which number am I?

**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

Which number, when 1 is subtracted from it, becomes a square number?

**The Lagrange Challenge**

*Lagrange’s Four-Square Theorem* states that every 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 **174**, in NINE different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using **4**, **6** and **10** once each, with + – × ÷ available, which THREE target numbers is it possible to make from the list below?

1 3 6 10 15 21 28 36 45 55 66

#*TriangularNumber*s

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

Can you arrive at **174** by inserting **9**, **10**, **16** and **20** into the gaps on each line?

- (◯+◯)×(◯–◯) = 174
- ◯×◯+◯–◯ = 174

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

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