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!