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

Consider all whole numbers from **1 to 60**, then delete the following:

- all prime numbers,
- … and any number that differs by 1 from a prime,
- all square numbers,
- … and any number that differs by 1 from a square,
- all multiples of 5,
- … and any number that differs by 1 from a multiple of 5,
- all multiples of 7,
- … and any number that differs by 1 from a multiple of 7.

One number will remain, what is it?

**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 & 7th rows contain the following fourteen numbers:

2 4 9 11 14 15 22 24 27 30 40 70 72 77

What is the difference between the highest prime number and highest square number?

**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 TWO ways of making **20 **when using *Lagrange’s Theorem*. Can you find both?

**The Mathematically Possible Challenge**

Using **5**, **6** and **8 **once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

11 22 33 44 55 66 77 88 99 110

#*11TimesTable*

**The Target Challenge**

Can you arrive at **20** by inserting **1**, **4**, **6** and **8** into the gaps on each line?

- (◯–◯)×(◯–◯) = 20
- (◯÷◯+◯)×◯ = 20
- (◯+◯)×◯–◯ = 20

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

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