### Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

### Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

# Back to the Planet of Vuvv

##### Age 11 to 14Challenge Level

Numbers are written by arranging digits in a row and each place in the row has a different value. This value depends on the base of the number system. The most common base nowadays is 10:

 10x10x10x10 10x10x10 10x10 10 1 Ten thousands Thousands Hundreds Tens Units/Ones

We often use these short forms for the columns:

 TTh Th H T U

To count in different bases, we just group numbers in a different way. For example, for base 2:

 2x2x2x2x2x2 2x2x2x2x2 2x2x2x2 2x2x2 2x2 2 1 Sixty fours Thirty twos Sixteens Eights Fours Twos Units/Ones

Zios count in base 3 so their numbers are grouped like this (we shall only look at the first three columns):

 3x3 3 1 Nines Threes Units/Ones

Let's work out what a Zio's 111 is in human numbers (base 10):

 Nines Threes Units/Ones 1 1 1

So, 111 = (1 x 9) + (1 x 3) + 1 = 13.

Zepts count in base 7 so their numbers are grouped like this:

 7x7 7 1 Forty nines Sevens Units/Ones

Let's see what a Zept's 111 is in base 10:

 Forty nines Sevens Units/Ones 1 1 1

So, 111 = (1 x 49) + (1 x 7) + 1 = 57.

To find out which way each type of creature is facing, calculate each number in human counting (base 10).

Remember that the creatures must be seeing numbers which could be a combination of Zios' and Zepts' legs.