Choose any three by three square of dates on a calendar page...
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some other
possibilities for yourself!
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
You can work out the number someone else is thinking of as
follows. Ask a friend to think of any natural number less than 100
but not to tell you which number they have in mind. Then ask them
to tell you the remainders when this number is divided by 3, when
it is divided by 5 and when it is divided by 7. Now multiply the
first remainder by 70, the second remainder by 21 and the third
remainder by 15 and add the three answers together. Now subtract
multiples of 105 from this total to give as small a positive whole
number as possible. This will be the number your friend first
thought of. Test this out a few times and explain why it works.