I had this idea this morning and was going to make another "figure-out-the-sequence" out of it, until it occurred to me that it was slightly more interesting than that. Consider the following procedure: start a line with 0 2, and add on additional numbers as you please so long as the following condition is met for each additional number: label the preceding two numbers from the one you are adding on a and b (in order) and the new number must be a - b, a + b, a / b, or a * b. Here are lines made by that procedure that end in the numbers 1-4 respectively: 0 2 2 1 (0 + 2 = 2, 2 / 2 = 1) 0 2 (no additional numbers added) 0 2 2 1 3 (0 + 2 = 3, 2 / 2 = 1, 2 + 1 = 3) 0 2 2 4 (0 + 2 = 2, 2 * 2 = 4) I ended it here because 5 is the first one that's a little bit harder. Puzzlers: 1. Find lines ending in numbers as high as you can (say, shoot for 20; I just did 11). 2. Given that the last number so far on a line is x, how can you append numbers so that the number -x is the last on the line? 3. Given that the last two numbers in a line are arbitrary x y, can you append numbers until the line ends in y x? (note: if the last two numbers are x y, you can append x - y but not y - x) 4. Given that the last number on a line is arbitrary x, can you append numbers until the line ends in x 1 or in 1 x? 5. Can a line be constructed that ends in any positive integer whatsoever? I do not know the answers to 3 and 4 at present. If, in thinking about this, you come up with other puzzlers about this type of sequence, feel free to add them to the thread.