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Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

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 14
Challenge 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.