Dear Nrich,
Here is my solution to the three neighbours investigation. I really enjoyed doing this investigation.
Maddie Carr (Year 6)
Nrich Task: Three Neighbours
If you take three numbers that are 'next door neighbours' when
you count, they are called consecutive numbers.
For example I took 1,2 and 3. Then I added them together and they equalled 6.
I noticed that when I added 2,3 and 4 they equalled 9, and then when I added 3,4 and 5
it equalled 12.
I noticed that when adding three consecutive numbers the answer is always divisible by three.
I also noticed that because I'm adding three consecutive numbers the answers increase
by three, so if I added four consecutive numbers the answers would probably increase by
four.
3 consecutive numbers
1+2+3=6
2+3+4=9
3+4+5=12
4 consecutive numbers
1+2+3+4=10
2+3+4+5=14
3+4+5+6=18
My prediction was right, if I added four consecutive numbers the answers increased by four
But I noticed that when adding four consecutive numbers the answers aren't divisible by four.
5 consecutive numbers
1+2+3+4+5=15
2+3+4+5+6=20
3+4+5+6+7=25
4+5+6+7+8=30
Then if I add five consecutive numbers it increases by five and is divisible by five.
I noticed now that possibly because 3 and 5 are prime numbers that the consecutive number sequences are divisible by that prime number.
Therefore now I'm going to try seven consecutive numbers.
7 consecutive numbers
1+2+3+4+5+6+7= 28
2+3+4+5+6+7+8= 35
3+4+5+6+7+8+9= 42
I have worked out a rule: that if the number of consecutive numbers in the sequence is a prime number then the result will be divisible by that prime number.