SOLUTION: Find the next number in the following sequence: 3,2,3,12,35,78_____

List the numbers vertically in a column:

3
2
3
12
35
78

We can't see any pattern for that, so for each
number except the bottom number 78, we find the difference 
between each number and the number just below it and
write it out to the right, starting with 2-3 = -1:

 3  -1   
 2   1   
 3   9  
12  23
35  43
78

We can't see any pattern for that 2nd column either, 
so for each number except the bottom number 43, we 
find the difference between each number and the number 
just below it and write it out to the right:

 3  -1   2
 2   1   8   
 3   9  14
12  23  20
35  43
78

Aha!  we can see a pattern 2+6=8, 8+6=14, 14+6 = 20

So the next term in 2, 8, 14, 20 is 20+6 = 26. So we
write 26 at the bottom of that column:

 3  -1   2
 2   1   8   
 3   9  14
12  23  20
35  43  26
78

Now we can work backwards, and see to write 26+43 = 69 below 43,
so that 69-43 would = 26.

  3  -1   2
  2   1   8   
  3   9  14
 12  23  20
 35  43  26
 78  69

Now we can work backwards, and see to write 26+43 = 69 below 43,
so that 69-43 would = 26.

  3  -1   2
  2   1   8   
  3   9  14
 12  23  20
 35  43  26
 78  69

Now we can work backwards, and see to write 69+78 = 147 below 78,
so that 147-78 would = 69.

  3  -1   2
  2   1   8   
  3   9  14
 12  23  20
 35  43  26
 78  69
147

So the next number is 147.

[Sometimes you have to go all the way to
a single number without seeing a pattern. If
so then you just rewrite that number again under
itself, and then work backwards to the first column.]

Edwin