# Puzzle of the Week for 4 January 1999: Hints

One of the most useful techniques for discovering patterns in sequences is the successive difference method. For example, suppose that we need to get the next term in the sequence:

1, 37, 62, 78, 87, 91, ...
The successive difference method involves subtracting each term from its successor to make a new sequence. In this example, we subtract 1 from 37, 37 from 62, 62 from 78, etc., to get sequence 2:
36, 25, 16, 9, 4, ...
We might recognize this sequence already, but what if we don't? In this case, we can try the successive difference method on sequence 2; we subtract 36 from 25, 25 from 16, etc., to get sequence 3:
-11, -9, -7, -5, ...
(If you don't recognize sequence 3, try the successive difference method one more time -- you can't miss in this case!)

Now that we have a sequence that we recognize, we can work backwards. The next term in sequence 3 should be -3. We add this to the last term in sequence 2 to get its next term: 4 + (-3) = 1. Now that we have the next term for sequence 2, we add it to the last term of the original sequence to get the next term of that sequence: 91 + 1 = 92.

The successive difference method works very well for many problems of this type, though not for all of them (as you will discover if you try it on the three sequences in this puzzle).