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What Numbers Can We Make?

Stage: 3 Challenge Level: Challenge Level:1
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.
This makes your overall answer 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!