Choose any three by three square of dates on a calendar page...
Can you explain how this card trick works?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
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:
We often use these short forms for the columns:
To count in different bases, we just group numbers in a
different way. For example, for base 2:
Zios count in base 3 so their numbers are grouped like this (we
shall only look at the first three columns):
Let's work out what a Zio's 111 is in human numbers (base
So, 111 = (1 x 9) + (1 x 3) + 1 = 13.
Zepts count in base 7 so their numbers are grouped like
Let's see what a Zept's 111 is in base 10:
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.