I discovered that all odd numbers except 1 could be made when you add two
consecutive numbers and to make any odd number all you had to do was
subtract 1 from the number then divide it by 2 then you have one of the
consecutive numbers and if you add 1 to it the new number is the 2nd number in
the sum. For example:11 -1 =10. 10 divided by 2 is five. So 5 and 6 are the
consecutive numbers that add up to 11.
I also discovered that all powers of two couldn't be made from any consecutive
numbers but all other even numbers could be from three or more consecutive
numbers. I made two discoveries. The first discovery I made is that there is a
pattern like this (10 = 1+2+3+4); (14 = 2+3+4+5) (18 = 3+4+5+6) (22 =
4+5+6+7). I also found that there is a sequence of 3 consecutive numbers that
add up to even numbers that are 6 apart: So if you start at 1, the first total you
get is 6 (1+2+3); then 12 (3+4+5) then 18 (5+6+7). A sequence of 4 consecutive
numbers that add up to even numbers that are 4 apart (10, 14, 28 etc). A
sequence of 5 consecutive numbers generates a sequence of even totals that is
10 apart (10, 30). There are no sequences of 6 even numbers that generate an
even number. A sequence of 7 consecutive numbers generates totals that are 14
apart (28, 42). I stopped at 8 because the first total is over 30.
If two numbers are consecutive one of them will be odd. Because an odd
and an even always make an odd number the result will be odd and never
even. To find out which numbers will work you just half the odd number
and the result will be one of the two numbers and the next consecutive
number will the other. What I discovered was that the general rule is that
if you take two consecutive numbers and then the next two consecutive
numbers etc etc the totals will be two apart. If you take a series of 3
numbers and total them the totals will be 3 apart. If you take a series of 4
numbers and total them the totals will be 4 apart etc etc. The reason why
you get a different pattern with the even numbers is that you can only use
sequences that have an even number of odd numbers in them and not
just any sequence. The reason that you cannot have any sequence of 6
consecutive numbers is that there will always be 3 odd numbers in a
sequence of 6 and that will always make an odd total.
Maia Webb
What would happen if you used minus
numbers? I did most of my search for
patterns starting at 1 but a next step would
be to see how it works if you start at other
numbers.