What Numbers Can We Make?

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:
 

 

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

Excellent. Thanks!