But not normal consecutive numbers. Not like 5678 or 4321.

They have to be out of order. For example, if you took the sequence 6789...

and scrambled it up, giving you something like 9687.

A game I like to play with myself while driving along a road is to predict when the next one will be. For example, if the address numbers are going up and I see 3109, then I have to find the lowest one which is drawn from the set 1234, that is higher than 3109. The answer, then, is 3124.

But here's a question: How many possible four digit non-sequential consecutive permutations are there (that's a long name; I need to come up with a better one)?

Well, first we need to figure out how many possible permutations there are of any given four digits.

If we take the simple 1234, then this, I think, is the total of all possible arrangements:

2134 2143 2314 2341 2413 2431

3124 3142 3214 3241 3412 3421

4123 4132 4213 4231 4312

That gives me 24 possible arrangements, and since I don't consider 1234 or 4321 to be good, then I think I can just go ahead and cross those out, giving me a grand total of 22.

But that's not all. In a base ten system, these are all the possible consecutive four digit combinations, each of which have 22 "valid" permutations:

1234 2345

3456 4567

5678 6789

There are six. And that multiplied by 22 is 132. 132 "good" street addresses in the four digit range.

But I'm not done yet - I'd like to look at the relationships between these numbers - including sequential ones, like 1234.

To find a full list of numbers, I will take the list of permutations of "1234" from further up, then change them to 2345, 3456, etc. by placing them in a text document, and find-and-replacing 1s with 2s, 2s with 3s, etc.

Before I do that, though, it's worth mentioning that the total of numbers with consecutive digits from 1000 to 9999, is 144. Interesting.

Now that I've done that, the first thing that I did was put all the numbers into numerical difference, and looked at the difference between each number, and the one next to it. It doesn't make a lot of sense.

Anyway, enough of that. Let's make a graph.

This is very curious. There's definitely a pattern, but what is it?

My theory is that what we're looking at is a fractal, so to speak. By that I mean, I think that every time the data points pass a power of ten and a digit returns to zero, there's a little jump in the graph. Also, I think that the larger the power of ten, the larger the jump in the line. In fact, if you will direct your attention towards the edges of the graph, I would assume that those are even larger "jumps", associated with 10,000. If you kept zooming out, I believe that there would continue to be larger and larger jumps with each new digit, on to infinity. And who knows what might happen there.