1. Spirolaterals
Starting with a number sequence, like 12345, and using graph paper, move 1 unit right, 2 units down, 3 units left, 4 units up, 5 units right, 1 unit down, 2 units left, and so on, until you begin retracing the pattern, or until it's obvious that this will never happen.
Try some of these sequences:
124
125
225
235
123
1234
12345
123456
112
1122
11223
112233

2. Try some sequences of your own
Try some sequences of your own. Make some observations about the resulting patterns. How does the number of digits in the sequence affect the pattern? How does changing a digit in a sequence change the pattern? Can you predict anything about which sequences will repeat and which will spiral off into space?