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?
If each point is joined to every other point then this makes (6
x 5)/2 = 15 line segments. All the line segments are coloured
either red or green.
Joel of ACS (Barker), Singapore proved
that there must then be a triangle all of whose sides are the same
colour. This is his solution:
Label the points from A to F. Out of the line segments connected
to A, at least 3 have to be the same colour, (let's say AB, AC and
AD are red). Then BC has to be green, because otherwise ABC will be
all red. Ditto for CD and BD. So BCD will be all green.