Place value

problem
Repeaters

Age

11 to 14

Challenge level

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.
problem
2-digit square

Age

14 to 16

Challenge level

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?
problem
Latin numbers

Age

14 to 16

Challenge level

Can you create a Latin Square from multiples of a six digit number?
problem
Basic rhythms

Age

16 to 18

Challenge level

Explore a number pattern which has the same symmetries in different bases.
problem
Binary squares

Age

16 to 18

Challenge level

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?
problem
Novemberish

Age

14 to 16

Challenge level

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.
problem
DOTS division

Age

14 to 16

Challenge level

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}.
problem
BT... eat your heart out

Age

16 to 18

Challenge level

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?
problem
Purr-fection

Age

16 to 18

Challenge level

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

Age

7 to 11

Challenge level

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?