### Calendar Capers

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!

### Rotating Triangle

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

# What Numbers Can We Make?

##### Stage: 3 Challenge Level:
Lots of people added together three numbers and noticed something special! For example, Emma, from Interlakes Elementary, told us her findings:

I figured out that, if you kept adding the numbers together, they will always come to a multiple of 3 every time you do it.

Charlie, from St. Cecilia's Wandsworth, noticed:

What we did was: we started with 3 numbers, then we added them together, and we noticed that some were odd and some were even but they were all multiples of 3. Then we tried adding 4 numbers, and we found that the answers were 1 more than the numbers in the 3 times table. Choosing 5 numbers we got answers which were 2 more than the 3 times table. We guessed that adding 6 numbers would give answers back in the 3 times table.

Great observations! Bethan, Gareth and Aditya, from St. Nicolas CE Junior School, offered an explanation of this. Bethan wrote:

When you have your three numbers, say 4, 7 and 1, each of these numbers is 1 more than a multiple of 3.
So 3+1, 6+1 and 0+1.
Then when you add them together, you can add the 3, 6 and 0 together which makes a multiple of 3, plus the three 1s left over will add together to also make a multiple of 3.

Good! (Is this related to either Charlie's or Alison's method?) Aditya used the same method to notice:

Adding together 99 numbers would give a multiple of 3, and 100 numbers would equal a multiple of 3 plus 1.

Brandyn from Garden International School considered what happened when he added sets of 3, 4, 5, 6 and 99 numbers from the bags below:

Choosing just three numbers from the bags above gave the following totals, all multiples of 3:

 1 4 7 10 TOTAL 3 3 3 12 3 21 3 30 2 1 6 2 1 9 2 1 12 1 2 9 2 1 15 2 1 18 1 2 15 1 2 18 2 1 24 1 2 21 1 2 24 1 2 27 1 1 1 12 1 1 1 15 1 1 1 18 1 1 1 21

Choosing four numbers from the bags above gave the following totals, all 1 more than (or 2 less than) multiples of 3:

 1 4 7 10 TOTAL 4 4 4 16 4 28 4 40 3 1 7 3 1 10 3 1 13 1 3 13 3 1 19 3 1 22 1 3 22 1 3 25 3 1 31 1 3 31 1 3 34 1 3 37 2 2 10 2 2 16 2 2 22 2 2 22 2 2 28 2 2 34 2 1 1 13 2 1 1 16 2 1 1 19 1 2 1 16 1 2 1 19 2 1 1 25 1 1 2 19 1 2 1 25 1 2 1 28 1 1 2 25 1 1 2 28 1 1 2 31 1 1 1 1 22

If I choose 5 numbers I predict that the series will start with 5 and increase in 3's.

If I choose 6 numbers I predict that the series will start with 6 and increase in 3's, etc..

If I choose 99 numbers, I predict that the series will start with 99 and increase in 3's.

Excellent. Thanks!