### 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?

# Take Three from Five

##### Stage: 3 and 4 Challenge Level:

This problem builds on What Numbers Can We Make?

Take a look at the video below. Will Charlie always find three numbers that add up to a multiple of 3?

If you can't see the video, click below to read a description.

Charlie invites James and Caroline to give him sets of five whole numbers. Each time he chooses three of their numbers that add together to make a multiple of 3:

 TOTAL 3 6 5 7 2 18 7 17 15 8 10 39 20 15 6 11 12 33 23 16 9 21 36 48 99 57 5 72 23 228 312 97 445 452 29 861 -1 -1 0 1 1 0

Charlie challenges Caroline and James to find a set of five whole numbers that doesn't include three that add up to a multiple of 3.

Can you come up with a set of five whole numbers that don't include a subset of three numbers that add up to a multiple of 3?

You can use the interactivity below to input sets of five numbers and test whether there are three numbers that add up to a multiple of 3.

This text is usually replaced by the Flash movie.

If you can't find a set of five whole numbers where it's impossible to choose three that add up to a multiple of three, convince us that no such set exists.