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

Following on from **DAY 187**, here’s another one of the unique *Keith Number* challenges. This was made famous by **Mike Keith** and if you like playing around with numbers, you’ll love having a go at this fun concept.

The **1st** 2-digit *Keith Number*, **14**, is worked out by following a pattern:

- 1+4=5; 4+5=9; 5+9=
**14**(the total arrives back to the original number).

The **2nd** 2-digit *Keith Number*, **19**, is worked out in a similar way:

- 1+9=10; 9+10=
**19**(again, the total arrives back to the original number).

By following this *Fibonacci*-style pattern, find the **3rd** and **4th** 2-digit *Keith Numbers*.

(Hint: both are less than 50)

**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 3rd & 6th rows of the playing board contain the following fourteen numbers:

5 12 13 18 20 25 33 36 42 45 49 56 66 80

Can you find three different numbers listed that add up to exactly 100?

**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 THREE different ways to make **240 **when using *Lagrange’s Theorem*. Can you find them?

**The Mathematically Possible Challenge**

Using **2**, **5** and **10 **once each, with + – × ÷ available, which are the only TWO numbers it is possible to make from the list below?

7 14 21 28 35 42 49 56 63 70

#*7TimesTable*

**The Target Challenge**

Can you arrive at **240** by inserting **1**, **2**, **3**, **4** and **5** into the gaps below?

- ◯²×◯×◯×(◯–◯) = 240

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

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