### Calendar Capers

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

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!

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

# Tri-colour

##### Stage: 3 Challenge Level:

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.