Place value

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

What is the smallest perfect square that ends with the four digits 9009?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers and so on?

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

Find b where 3723(base 10) = 123(base b).

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

Can you find the chosen number from the grid using the clues?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.