"9, 12 and 15 can all be written using three consecutive numbers. I wonder if all multiples of 3 can be written in this way?"
All numbers can be written in this way. The equation for the multiples of 3 would be y=(x-1) + (x) + (x+1). Y is the multiple of 3. Y divided 3 = x
ex. y=150
y divided 3 =x
x =50
x+1 =51
x-1 =49
y=(x) + (x+1) + (x-1)
= 50 + 51 + 49
=150
49, 50 and 51 are all consecutive numbers and the sum of the 3 numbers is a multiple of 3(150).
ex.2 y=210
x =70
x+1 =71
x-1 =69
70 +71 +69 =210
ex.3 y=423
x =141
x+1 =142
x-1 =140
141+142+140 =423
"Maybe you could write the multiples of 4 if you used four consecutive numbers..."
No this would not work. An example of this would be 16.
There isn't a way to get 16 with 4 consecutive numbers. The equation used for multiples of 3 with 3 consecutive numbers would not work because there are an even nnumber of numbers. The formula would work for a multiple of 5 with 5 consecutive numbers. The example is:
y=200
x =40
x-1 =39
x-2 =38
x+1 =41
x+2 =42
38+39+40+41+42 =400.
This formula works because 5 is an odd number and the formula only works for an odd amount of numbers with an odd multiple.
"I wonder if we could write every number as the sum of consecutive numbers?"
This is again, not true. Even numbers that are not are multiple of an odd number like 16. Like I mentioned, this is because the formula does not work with even numbers. The equation only works with odd numbers and odd multiples.
"Some numbers can be written in more than one way! I wonder which ones?"
There are many numbers that can be written with more than one consecutive pattern. An example of this would be 15, which can be written in 3 consecutive patterns.
7+8
4+5+6
1+2+3+4+5
Another example of this would be 9.
4+5
2+3+4