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

Fill the 15 gaps below with the numbers **1-15**, once each, so all five lines work out:

◯ + ◯ = ◯

◯ + ◯ = ◯

◯ + ◯ = ◯

◯ + ◯ = ◯

◯ + ◯ = ◯

The concept behind this challenge is similar to my **Mathelona** number puzzles, so please feel free to click the link for details of my popular pocket book of challenges.

**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 4th & 6th rows contain the following fourteen numbers:

3 5 10 12 18 20 32 33 35 44 49 54 56 60

Which 2-digit number, when 4 is subtracted from it, becomes a multiple of 9?

**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 **163**, in SEVEN different ways, when using *Lagrange’s Theorem*.

**The Mathematically Possible Challenge**

Using the three digits **3**, **5** and **8** once each, with + – × ÷ available, which are the only FOUR numbers it’s possible to make from the list below?

1 3 6 10 15 21 28 36 45 55 66

#*TriangularNumbers*

**The Target Challenge**

Can you arrive at **163** by inserting **5**, **7**, **9** and **20** into the gaps on each line?

- ◯×◯+◯×◯ = 163
- ◯×◯+◯+double◯ = 163

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

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